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FOR  S ECO^Mf^  SGH 00 IJ 

THURSTON 


CMILLAN'S 
OMMERCIAL 


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BUSINESS  ARITHMETIC 


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SECONDAEY  SCHOOLS 


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BUSINESS   AEITHMETIC 


FOR 


SECONDARY  SCHOOLS 


BT 

ERNEST  L.  THURSTON 

SUPEBINTBNDENT   OF    PLBLIC    SCHOOLS,    DISTRICT   OF    COLUMBIA 


"^  ^•>  .    «  ' 


Neto  ¥0rlt 
THE  MACMILLAN  COMPANY 

LONDON:   MACMILLAN  &  CO.,  Ltd. 

1917 


Copyright,  1913, 
By  the  MACMILLAN  COMPANY. 


Set  up  and  electrotyped.     Published  March,  1913.     Reprinted 
August,  September,  igi3;  April,  August,  1914;  February,  1916; 
August,  1917.       •      • 


*  t  •  •  •  • 


Norteooti  ^rtjjji: 
Berwick  &  Smith  Co.,  Norwood.  Mass.,  U.S.A. 


PREFACE. 

The  use  and  applications  of  number  vary  with  the  changes 
and  development  of  our  industrial,  commercial  and  social 
relations.  The  arithmetic  of  to-day  is  not  the  arithmetic  of 
ten  years  ago.  The  fundamental  principles  and  processes  do 
not  alter,  but  new  applications  constantly  develop  and  older 
applications  lessen  in  usefulness.  In  this  sense,  arithmetic  is 
a  living  subject  which  deals  with  living  phenomena. 

In  these  days  arithmetic  is  commonly  used  not  only  for  the 
necessary  routine  computations  incident  to  our  private,  busi- 
ness, or  scientific  affairs,  but  its  processes  are  called  upon  to 
study  facts  through  number  and  to  interpret  these  facts — to 
put  number  into  things  rather  than  to  take  number  from  them. 
Moreover,  the  modern  use  of  number  often  takes  the  form  of 
a  language,  as  in  the  construction  or  reading  of  numerical 
illustrations,  and  statements  or  statistics  which  are  designed 
to  impress  upon  others  certain  facts  or  conditions.  Thus 
modern  arithmetic  may  be  said  to  have  a  use  as  a  tool,  i.  e., 
for  routine  computation,  an  interpretive  use,  and  a  language 
use. 

But  accuracy  and  facility  in  the  use  of  most  working  instru- 
ments presuppose  a  knowledge  of  the  fundamental  parts  of 
the  instrument,  of  the  interrelation  of  these  parts,  of  their 
working  principles,  and  of  the  classes  of  material  or  work  to 
which  they  may  be  applied.  Thus,  facility  in  the  three  "  uses  '* 
of  the  arithmetic  instrument  is  founded,  primarily,  on  a  logical 
working  knowledge  of  the  fundamental  operations  and  prin- 
ciples. This  is  the  more  necessary  because  the  arithmetic 
problems  of  actual  life  do  not  have  a  text-book  form  of  state- 
ment. If  formulated  at  all,  they  may  offer  no  direct  clue 
to  a  particular  subdivision  of  the  subject.  Often  the  problem 
must  first  be  formulated,   at  least  mentally,  the  power  of 

4f?9940 


VI  PREFACE. 

selection  being  exercised  to  choose  from  a  group  of  number 
facts  those  essential  to  the  computation  desired. 

This  book  is  an  attempt  to  construct  the  arithmetic  instru- 
ment logically,  in  its  simplest  form,  as  a  branch  of  mathematics; 
then  to  apply  this  tool  to  general  computation,  and  to  the 
study,  preparation  and  interpretation  of  varied  number 
material.  Involved  in  this  is  the  purpose  to  develop  arith- 
metic as  a  language  of  business,  or  as  a  means  of  interpretation 
and  study  of  business,  and  economic  conditions;  to  acquire 
the  capacity  properly  to  present  numerical  facts  by  tabulations 
or  graphs;  to  cultivate  clearness  of  thought  and  expression, 
and  an  appreciation  of  the  value  of  order  and  system  in  applied 
number  work. 

In  the  preparation  of  the  subject  matter  of  this  book,  no 
attempt  has  been  made  to  draw  from  other  text-books,  al- 
though much  may  be  found  here  that  is  in  harmony  with 
them.  Material  has  been  gathered  only  from  living  and 
reliable  sources.  The  material  has  been  selected  with  the 
purpose  of  emphasizing  how  arithmetic  is  applied  for  a  definite 
object.  The  problem  exercises  are  not  intended,  as  is  so 
often  the  case,  as  a  means  of  conveying  to  the  mind  indigestible 
information  on  most  known  subjects.  The  author  believes 
that  the  book  will  be  found  none  the  less  rich  in  content  because 
it  aims  at  the  digestion  of  facts,  rather  than  at  surfeiting 
with  them. 

Much  thought  has  been  given  to  the  problem  work  and, 
as  one  result,  considerable  variety  will  be  noticed  in  form  of 
statement.  Thus  the  old  time  "problem  in  question  form" 
is  used,  also  the  incomplete  statement,  the  statistical  table, 
the  business  form,  the  written  article,  the  memorandum,  etc. 
Place  is  also  given  to  series  or  related  problems,  composite 
problems,  and  "central"  problems,  to  be  viewed  from  different 
standpoints.  Exercises  in  the  preparation  of  original  problems 
are  included  for  the  purpose  of  cultivating  powers  of  selection 


PREFACE.  vii 

and  statement,  and  of  distinguishing  between  essentials  and 
non-essentials.  Mental  development,  and  growth  in  power  of 
analysis,  must  necessarily  follow  the  doing  of  the  work  that 
these  exercises  require. 

A  method  of  development  has  been  followed,  in  condensed 
and  simplified  form,  and  rules  have  been  reduced  to  an  absolute 
minimum.  It  has  been  assumed,  however,  that  the  pupil  has 
had  considerable  experience  in  grade  arithmetic,  before  under- 
taking the  study  of  this  book.  For  this  reason  illustrative 
examples  are  frequently  given  only  in  outline.  To  give 
facility  in  the  fundamental  operations  and  main  processes, 
a  considerable  quantity  of  abstract  work  has  been  introduced. 
In  this  line,  special  emphasis  has  been  laid  on  the  fundamental 
processes  as  applied  to  whole  numbers,  decimals  and  fractions. 
Short  methods  have  been  emphasized  for  paper  work,  and 
methods  of  checking  have  been  introduced  or  suggested.  It  is 
felt  that  every  important  computation  should  be  checked. 

Oral  exercises,  planned  with  the  utmost  care  to  serve  as 
models,  have  been  given  throughout  the  book,  but  they  should 
be  supplemented  by  further  work  along  the  lines  that  they 
suggest.  The  exercises  should  vary  in  form,  being,  at  dif- 
ferent times,  sight  exercises,  exercises  by  dictation,  and  exer- 
cises based  upon  some  fundamental  expressions  that  are 
placed  upon  the  board,  and  around  which  individual  examples 
are  built.  Many  types  of  exercises  will  be  found  illustrated  in 
the  following  pages. 

In  all  oral  work,  it  should  be  remembered  that  facility  will 
come  (1)  from  constantly  varying  the  simple  number  com- 
binations, or  "ringing  the  changes"  on  fundamentals,  and 
(2)  from  fitting  the  examples  to  the  individual  pupil,  giving 
him  something  that  he  can  do  and  that  will  develop  him  in  the 
doing.  Attention  to  these  requirements  will  bring  a  rich 
return  in  the  increased  capacity  of  the  pupils. 

The  author  wishes  to  acknowledge  his  indebtedness  to  Mr. 


viu  PREFACE. 

Charles  Hart,  head  of  department  of  business  practice,  Busi- 
ness High  School,  Washington,  D.  C,  who  read  the  first  proof, 
and  whose  wise  counsel  and  careful  criticism  have  been  helpful. 

E.    L.    T. 
Washington,  D.  C. 


EDITOR'S  INTRODUCTION. 

No  subject  of  the  curriculum  has  received  recently  more 
discussion  and  criticism  than  has  arithmetic.  In  that  most 
suggestive  book  "School  and  Society''  John  Dewey  dwells  on 
the  facts  that  there  should  be  established  a  natural  connection 
of  the  everyday  life  of  the  child  with  "the  business  environ- 
ment of  the  world  around  him,"  and  "it  is  the  affair  of  the 
school  to  clarify  and  liberalize  this  connection."  Observation 
confirms  the  dictum  of  John  Stuart  Mill  that  the  child  is  by 
nature  logical — he  seeks  the  reasons  for  things.  The  teaching 
of  mathematics  as  an  abstract  science  based  on  phenomena 
outside  of  the  child's  experience  serves  to  dull  his  natural 
logical  habit  of  mind. 

The  three-fold  aim  of  teaching  arithmetic  is  now  generally 
conceded,  and  a  new  text  can  scarcely  fail  to  take  these  into 
account.  First,  it  should  give  the  ability  to  compute  correctly 
and  with  some  rapidity.  Second,  it  should  sustain  the  interest 
of  the  learner,  that  is,  arithmetic  should  present  a  body  of 
knowledge  that  is  fresh  and  attractive.  Third,  the  teaching 
of  arithmetic  should  develop  the  power  of  applying  number 
concepts  to  the  activities  of  everyday  life. 

The  statement  of  these  ideals  is  easy:  the  difficulties  come 
in  their  realization.  Arithmetic  can  be  made  an  instrument 
of  education,  both  for  what  has  commonly  been  termed  culture 
and  for  practical  affairs.  No  school  study  offers  better  oppor- 
tunities for  teaching  pupils  to  think  or  for  directing  their 
thinking  to  the  sort  of  phenomena  with  which  they  will  deal 
when  school  is  at  an  end.  The  author  and  the  editor  of  this 
book  recognized  these  ideals  when  the  book  was  planned, 
eight  years  ago.     In  preparing  the  book  certain  fixed  prin- 

ix 


X  EDITOR'S  INTRODUCTION. 

ciples  have  been  borne  in  mind.  One  is  Herbert  Spencer's — 
that  "so  far  as  possible  the  child  should  make  his  own  in- 
vestigations and  draw  his  own  conclusions."  Constantly  in 
this  book  the  pupil  is  asked  to  investigate  for  himself,  to  for- 
mulate or  arrange  his  data,  and  to  deduce  his  own  conclusions. 

Secondly,  this  book  is  built  on  the  assumption  that  through 
the  use  of  concrete  problems,  it  is  possible  to  do  three  things 
at  once,  namely,  to  give  drill,  to  keep  up  interest,  and  to  train 
in  the  facility  of  practical  application.  In  the  selection  of 
material  little  or  no  regard  has  been  given  to  text-books  here- 
tofore existing.  Problems  have  been  drawn  almost  entirely 
from  home,  store,  farm,  shop,  the  engineer's  office,  and  other 
like  sources. 

We  have  progressed  too  far  educationally  to  spend  time  in 
arguing  that  it  is  saner  to  teach  pupils  to  think  on  the  data 
that  they  will  later  use,  than  to  try  to  teach  them  from  the 
obsolete  data  of  the  books.  Not  only  does  the  first  method 
arouse  more  interest;  it  gives  the  only  assurance  that  the 
pupils  will  be  trained  for  practical  affairs. 

The  author  would  seem  to  have  had  an  almost  ideal  prep- 
aration for  the  writing  of  this  book.  In  the  first  place  he 
was  broadly  trained  in  mathematics,  and  for  several  years 
had  experience  as  a  teacher  in  an  institution  of  higher  learning. 
In  addition  to  this  he  had  some  years  experience  as  a  teacher 
of  arithmetic  in  the  Business  High  School  at  Washington, 
and  more  recently  has  served  as  a  supervisor  in  methods  of 
teaching  in  elementary  schools.  Best  of  all  he  is  a  trained 
and  careful  investigator,  who  has  been  willing  to  forsake  the 
traditions  of  arithmetic  to  work  out  by  practical  investigation 
a  book  along  new  lines. 

This  book  gives  but  limited  attention  to  the  fundamental 
operations,  and  the  attention  that  is  given  is  only  to  emphasize 
the  relative  value  of  the  fundamental  processes.  As  with  the 
other  book  of  the  series,  decimals  are  made  to   follow  im- 


EDITOR'S   INTRODUCTION.  xi 

mediately  after  whole  numbers,  thus  combining  these  closely 
related  parts  of  our  notation. 

The  difficulty  with  much  arithmetic  teaching  is  that  the 
pupil  considers  arithmetic  entirely  a  matter  of  the  book, 
having  nothing  to  do  with  practical  affairs.  The  pupil  who 
works  through  this  book  should  have  an  intelligent  notion  of 
the  modern  organization  and  operation  of  business,  a  grasp 
of  himself  and  arithmetical  processes,  and  such  facility  in  the 
application  of  these  processes  as  will  enable  him  to  go  out  and 
make  himself  useful  in  the  work-a-day  world. 

C.  A.  H. 

GiRARD  College, 
January  1,  1913. 


TABLE  OF  CONTENTS. 


Chapter  Pagb 

I.     Units  and  Numbers 1 

II.     Notation  and  Numeration 3 

III.  Addition .  . 7 

IV.  Subtraction 18 

V.     Multiplication 25 

VI.     Division 35 

VII.     Arithmetical  Averaging 43 

VIII.     The  Equation 46 

IX.     United  States  Money 49 

X.     Making  Change 54 

XL     Postage 56 

XII.     Payment  for  Services 61 

XIII.  Business  Terms  and  Accounts 72 

XIV.  Advertising 77 

XV.     Factors  and  Multiples 86 

XVI.     Reduction  of  Fractions 91 

XVII.  Fractions  —  Fundamental  Processes  —  Con- 
version   « 95 

XVIII.     Fractional  Relations  of  Numbers 106 

XIX.     Aliquot  Parts 110 

XX.     Problem  Analysis  and  Solution 118 

XXI.     Involution  and  Evolution 121 

XXII.     Denominate  Numbers 132 

XXIII.     Practical  Measurements 143 

XXIV.     Measurement  of  Time '. 164 

XXV.  Practical     Measurements  —  Temperature, 

Composite  Units,  Formulae 175 

XXVI.     The  Metric  System 182 

xiii 


XIV 


TABLE  OF  CONTENTS. 


Chapter  Page 

XXVII.     Ratio  and  Proportion 186 

XXVIII .  Graphic      Arithmetic  —  Scales,      Plotting, 

Graphs 193 

XXIX.     Percentage 213 

XXX.     Profit  and  Loss 225 

XXXI.     Commercial  Discount 236 

XXXII.     Agency 245 

XXXIII.  Insurance  —  Personal  and  Property 255 

XXXIV.  Taxation     and     Public     Revenue  —  Local, 

State  and  National 269 

XXXV.     Interest  —  Simple 285 

XXXVI.     Interest  —  Compound 302 

XXXVII.     Loans  and  Payments 306 

XXXVIII.     Savings  Accounts 326 

XXXIX.     Stocks  and  Bonds 332 

XL.     Domestic  and  Foreign  Exchange 355 

XLI.     Depreciation 375 

XLII.     Cost-Keeping 379 

XLIII.     Bids  and  Estimates 383 

XLIV.     Partitive  Proportion  and  Partnership 387 

XLV.  Equation  of  Payments  and  of  Accounts. . . .  397 

XLVI.     Billing 403 

XLVII.     Storage. .  .^ 414 

Appendix 

I.     Signs  and  Symbols 419 

II.     Standard  Abbreviations 420 


BUSINESS  ARITHMETIC 

FOR 

SECONDARY  SCHOOLS 


BUSINESS  AEITHMETIO 

FOE  SECONDAEY  SCHOOLS 


CHAPTER  I. 

UNITS  AND  NUMBERS. 

In  applied  arithmetic,  constant  use  must  be  made  of  units 
of  value  and  of  measure.  Unity  means  one-ness,  but  a  unit 
is  not  necessarily  "one,"  or  a  single  thing,  for  it  may  be  any 
quantity  definitely  measured  or  stated.  One  dollar  is  a  unit, 
but  the  dollar  is  one  hundred  cents,  or  a  certain  number  of 
unit  grains  of  gold  and  alloy,  that  is,  a  collection  of  other 
units.  In  practice,  units  vary  with  the  things  with  which 
they  are  associated.  A  retailer  sells  goods  by  the  can — the 
can  is  his  selling  unit;  but  he  buys  by  the  case,  say  of  24  cans, 
and  the  case  may  be  termed  his  buying  unit.  The  idea  of  a 
unit  as  a  definite  quantity  is  emphasized  in  many  occupations, 
as,  for  example,  in  engineering,  where  a  unit  distance  is  often 
taken  as  100  or  1000  feet. 

It  is  evident  that  all  units  are  divisible.  If  the  unit  is  one 
or  a  quantity  considered  as  one,  it  is  termed  an  integral  unit. 
If  an  integral  unit  is  divided  into  tenths,  hundredths,  etc., 
one  of  the  parts  so  obtained  is  termed  a  decimal  unit.  If  the 
integral  unit  is  divided  into  any  number  of  equal  parts,  it  is 
termed  a  fractional  unit. 

A  number  is  a  unit  or  a  collection  of  units.  It  is  therefore 
integral,  decimal  or  fractional  according  to  the  character  of 
units  of  which  it  is  composed.  If  the  number  is  entirely  unaf- 
fected by  association  with  any  object  or  expression  it  is  an 
2  1 


2         •  BtSINESS  ARITHMETIC. 

ahstr'act  ''riurrvber:  '  (]6±ainples :  87,  34-)  If  the  number  is 
associated  with  some  idea  it  is  concrete.  (Example,  8  foxes.) 
If  the  associated  expressions  are  those  of  standard  measures, 
the  number  is  denominate.     (Example,  12  yards.) 

Numbers  are  like  or  unlike,  according  as  they  have  the 
same  or  different  unit  values.  Thus  "five  months"  and 
"eight  months"  are  like  numbers,  while  "Rve  months"  and 
"eight  pounds"  are  unlike. 

Numbers  are  simple  if  they  contain  units  of  the  same  kind; 
they  are  compound  if  they  contain  different  forms  of  units 
which  are  reducible  to  a  single  form  of  unit.  Thus  "4  lb." 
is  simple,  but  "4  lb.  8  oz."  is  a  compound  number  which  may 
be  reduced  to  the  simple  number  "72  oz." 

EXERCISE. 

1.  Name  six  units  of  measure. 

2.  Name  five  units  employed  in  different  trades  or  professions. 

3.  Classify  these  numbers  and  name  the  unit  in  each  case: 

(a)  976.  (d)  $3.85.  (g)  53^  gal. 

(6)  .08.  (e)  5  yd.  2  ft.  (h)  U  lb. 

(c)  45  lb.         (/)  276  payments.  (i)  4  horses,  2  mules. 

4.  Name  and  classify  the  numbers  and  units  in  these  expressions: 

(a)  "The  cost  of  this  cloth  is  55  cents  per  yard." 

(6)  "The  goods  are  packed  in  cases  of  48  boxes  each." 

(c)  "The  temperature  reached  96°  to-day." 

id)  "  It  will  take  at  least  half-an-hour  and  probably  forty  minutes." 

(e)  240  is  H  of  480. 


CHAPTER  II. 
NOTATION  AND  NUMERATION. 

INTRODUCTORY    EXERCISE. 

1.  How  do  the  expressions  "3  yd."  and  ".03  yd."  differ  in  meaning? 

2.  Add  ciphers  to  the  numbers  "40"  and  ".04"  in  such  a  way  as  not 
to  alter  their  values. 

3.  Where  does  the  insertion  of  a  cipher  cause  a  change  of  value? 

4.  In  expressing  numbers  orally,  what  are  the  values  of  the  syllables 
"teen"  and  "ty"? 

Arabic  Notation. 

The  Arabic  or  Hindu  notation  is  the  system  universally 
used  in  civilized  countries.  The  significant  figures  or  digits, 
1  to  9,  were  in  use  in  India  2,000  years  ago,  but  calculations 
were  performed  with  extreme  difficulty  until  the  invention 
of  the  tenth  symbol,  the  cipher  or  zero,  800  years  later.  The 
zero  made  modern  computation  possible.  In  the  seventeenth 
century,  the  invention  of  the  decimal  point  made  possible 
the  present  varied  use  and  application  of  number.  The 
present  tendency  is  toward  a  continually  broadening  use  of 
decimals. 

The  "significant"  figures  of  the  system  are  those  that 
have  a  name  or  face  value,  and  a  position  value.  The  zero 
serves  only  to  alter  the  position  value  of  other  figures. 

PRINCIPLES    OF   USE. 

1.  The  name  value  of  each  significant  figure  'm  constant. 

2.  The  position  value  increases  in  ten-fold  ratio  from  right  to  left. 
Note.     Hence  called  the  "decimal  system,"  from  the  I^atin  decern 

meaning  "ten." 

3.  A  decimal  point,  or  period,  separates  units  and  tenths  of  imits. 
Note.     The  portion  of  the  number  to  the  left  of  the  decimal  point  is 

termed  the  whole  number  or  integer;  the  portion  to  the  right  is  the  decimal. 

3 


4  BUSINESS   ARITHMETIC. 

4.  The  zero  increases  the  position  of  significant  figures  to  its  left  ten 
fold;  it  decreases  the  value  of  significant  figures  in  a  decimal,  to  its  right, 
to  one-tenth  their  previous  value. 

5.  In  any  number,  where  a  significant  intermediate  figure  is  lacking, 
a  zero  is  written  in  its  position. 

6.  Place  values  are  named  under  the  French  or  English  system. 
Under  the  French  system,  used  in  this  country,  the  figures  of  large  numbers 
are  divided  into  groups  or  periods  of  three  figures  each,  as  follows: 

Place  131211109  8  7       654  321       .        123  456 


Name 


g  a 


fl  2 


iSl    Jl_   211    ^      ^     II      Um 


C^Srs      !=^c3li:        SS^  J^sa       a».       o  ^,C  S^'^i 

H  WH«   «h:^     teHH       wh;i3    q     hwh        hw^ 

Period    BilUons  Millions  Thousands  Hundreds      Thousandths  Millionths 

Illustration  of  Numeration.    The  numbers  should  be  read  in  the 
simplest  way,  the  word  "and,"  or  "decimal,"  being  used  for  the  decimal 
point,  thus: 
14,264.216  is  read:  Fourteen  thousand,  two  hundred,  sixty-four  and  two 

himdred  sixteen  thousandths. 
2,132.       is  read:  Twenty-one  hundred,  thirty-two. 
3.02    is  read:  Three  and  two  hundredths. 
30,000.003  is  read:  Thirty  thousand  and  three  thousandths. 

Note.  In  English  tables,  the  periods  contain  six  places  each,  called 
units,  millions,  etc.  Under  this  system  the  number  36,000,000,000  would 
be  read:  Thirty-six  thousand  milHons. 

EXERCISE. 

1.  Which  is  the  larger,  3  yd.  or  .030  yd.? 

2.  How  does  the  decimal  point  determine  the  size  of  the  part  taken? 

3.  Read:  4.    Read:  6.    Read: 

52,608  2.43  30.0009 

1,292,723  17.061  2.0703125 

329,624,006  228.0406  .627724 

50,007,064,329  5,060.38001  2.390 

6.    Read  by  both  English  and  French  systems: 
472,693,684,721,125.001. 


NOTATION   AND   NUMERATION.  5 

7.  Insert  a  cipher  in  the  number  17,246  in  five  different  positions  in 
turn,  and  read  the  results.  In  which  position  has  the  cipher  the  least 
effect?    The  greatest? 

8.  Insert  a  cipher,  in  turn,  between  each  two  significant  figures  in 
the  number  86.043201,  and  read  the  resulting  numbers. 

9.  Read  .4  as  hundredths;  .65  as  ten-thousandths;  .025  as  hundredths; 
.002  as  millionths. 

10.  What  is  the  decimal  of  lowest  value,  that  can  be  written  with  the 
figures  3  and  7  and  foiu-  ciphers? 

Note.  In  writing  large  numbers,  separate  periods  by  commas,  or  by 
spacing. 

11.  Write  in  colunm:  Seven  billion,  two  hundred  forty-eight  million, 
seventy  thousand,  eight  hundred  twenty-eight;  two  hundred  twenty-six 
million,  four  hundred  thousand,  thirty;  eighteen  hundred  thousand,  four; 
six  ten-thousandths;  five  and  four  milUonths;  sixty-three  thousand  and 
six  hundred  thirty  hundred-thousandths;  four  hundred  fifty-nine  and  eight 
hundred  eleven  ten-thousandths;  fifty-four  tenths;  fiftj^-two  hundred- 
thousandths. 

12.  Write,  substituting  numbers  for  numerical  words:  (a)  The  cir- 
cumference of  a  circle  is  three  and  fourteen  thousand,  one  hundred  fifty- 
nine  hundred-thousandths  times  the  diameter.  (6)  A  nautical  mile  is 
one  and  one  hundred  fifty-two  thousandths  of  a  statute  mile,  (c)  The 
total  elongation  of  a  bar  on  a  steel  bridge,  under  a  maximum  load,  was 
two  hundred  fifty-six  thousandths  inches. 

Roman  Notation. 
The   Roman   system   employs   as   number  symbols   seven 
letters  of  the  Roman  alphabet.      These  characters  have  a 
name  value  and  a  variable  position  value. 

I    V    X     L      C        DM 

1  5  10  50  100  500  1000 

PRINCIPLES    OP   USE. 

1.  Combinations  of  symbols  are  written  from  left  to  right  in  order  of 
value.    The  total  value  is  the  sum  of  the  constituent  values. 

Illustrations.  CX  =  100  +  10  =  110.  MDCC  =  1000  +  500 
-f  100  +  100  =  1700. 


6  BUSINESS   ARITHMETIC. 

2.  To  avoid  the  consecutive  use  of  symbols  four  times,  a  letter  of 
less  value  may  be  written  before  one  of  greater  value,  the  combination 
being  equal  to  the  difference  of  the  name  values. 

Illustrations.     L  =  50,  but  XL  =  50  -  10  =  40.     IX  =  10  -  1  =  9., 

3.  The  mark  " — "  over  a  symbol  multiplies  its  value  by  1000. 

Illustration.    XIV  =  14,  but  XIV  =  14,000. 

EXERCISE. 

1.  Write  the  Roman  numerals  from  1  to  100. 

2.  Spell  the  Roman  numerals  from  1854  to  1913. 

3.  The  corner  stone  of  a  pubUc  building  is  marked  MDCCCLXVIII. 
It  was  laid  in  what  year  ?. 

4.  Write  in  Arabic  notation:  XIV;  XCVI;  MDVIII;  MCDIX: 
XXlfl;  CX;  DCXII. 

5.  Name  three  common  uses  of  the  Roman  notation. 

6.  Name  objections  to  the  general  use  of  the  system. 


CHAPTER  III. 

ADDITION. 

The  fundamental  operations  on  which  the  working  methods 
of  arithmetic  are  based  are:  (1)  addition;  (2)  subtraction; 
(3)  multiplication;  and  (4)  division.  These  processes  are 
applied  to  whole  numbers,  decimal  fractions  and  common 
fractions.  They  will  be  considered,  first,  in  their  relation  to 
whole  numbers  and  decimals. 

Addition  is  the  process  of  combining  two  or  more  numbers 
into  a  single  equivalent  number,  called  the  sum,  the  amount, 
or  the  total.  It  is  the  most  universally  employed  fundamental 
process,  and  should  be  performed  with  absolute  accuracy 
and  reasonable  speed.  From  a  business  standpoint,  this 
accuracy  and  speed  depend  on:  (1)  careful  formation  of 
figures;  (2)  alignment  of  figures;  (3)  capacity  to  give  and  take 
number  dictation;  (4)  capacity  to  copy  figures  accurately; 
(5)  working  knowledge  of  "grouping";  and  (6)  continued, 
systematic  practice. 

Careful  Writing.  Figures  should  be  uniform  in  size,  legible, 
simple  and  distinct  in  form,  evenly  aligned  vertically  and 
horizontally,  and  closely  spaced  but  not  crowded. 

EXERCISE. 

1.  Give  reasons  for  the  requirements  just  stated. 

2.  How  do  carelessly  formed  figures  reduce  speed  in  addition? 

3.  Write  an  original  column  of  6-place  figures.  (Exchange  papers 
and  criticise  for  form.) 

4.  The  money  column  on  page  331  is  "unit  ruled."  What  is  unit 
ruling?    Of  what  value  is  it  in  addition? 

Number  Dictation  is  frequently  incidental  and  preparatory 
to  addition.     One  clerk  may  dictate  values  to  another,  who 

7 


8  BUSINESS  ARITHMETIC. 

writes  them  in  column,  or  "checks  off"  a  list  he  holds.  In 
dictation,  numbers  must  be  clearly  enunciated  in  an  even 
tone,  without  unnecessary  words,  and  at  uniform  speed. 

SUGGESTION    FOR   TEACHERS. 

1.  Dictate  a  column  of  figures.  Have  pupils  check  accuracy  by 
"dictating  back." 

2.  Let  pupils  dictate  original  columns  to  each  other.  Later,  check 
by  addition. 

Copying  is  more  common  than  dictating.  Office  clerks 
are  continually  transferring  figures  from  one  record  to  another. 
Thus  a  bank  clerk  lists  a  depositor's  checks,  or  a  book-keeper 
posts  from  a  Cash  Book  to  a  Ledger. 

SUGGESTION    FOR    TEACHERS. 

1.  Have  the  class  copy  a  column  of  blackboard  figures.  Check  by 
dictation. 

2.  Exchange  original  columns  for  copjring. 

General  Addition. 
The  mental  process  of  adding  consists  in  grouping  digits 
of  the  same  order.  The  thought  should  be  placed  on  the 
result.  In  adding  5,  8,  9  and  7,  think  of  each  sub-total  (13, 
22,  29).  Facility  in  addition  is  gained  by  reading  instantly  any 
two  or  three  digits  as  a  sum.  Thus,  "8  +  9"  should  be  read 
"17." 

DRILL    TABLE. 

The  forty-five  two-figure  combinations:  Name  sums  at  sight. 
74241343314221189856455 
76537623251213199851434 


7 

1 

5 

6 

6 

8 

9 

8 

7 

7 

4 

9 

'  7 

6 

7 

5 

3 

2 

4 

5 

7 

6 

2 

8 

6 

6 

9 

6 

1 

2 

3 

5 

8 

3 

8 

7 

9 

9 

8 

9 

9 

8 

4 

2 

ADDITION. 


EXERCISE. 


1.  Name  the  totals.  4  +  8,  6  +  9,  3  +  1,  4  +  7,  2  +  8,  1  +  7,  3  +  9, 

4  +  5,  4  +  4,  14-2  +  3,  2+4  +  3,  5  +  2  +  7,  6  +  7+3,  8  +  2+4, 
1  +  1+3+2,  1+8  +  8. 

2.  List  all  possible  groups  of  two  or  three  digits  equaling  ten.  Leam 
them. 

3.  Develop  systematically  a  list  of  the  165  possible  combinations  of 
three  digits. 

Illustration.     1  +  1  +  1,  1  +  1+2,  1  +  1  +  3,  etc. 

4.  Give,  orally,  the  consecutive  totals  obtamed  by  adding:  3s  to  58 
(58,  61,  64,  etc.),  4s  to  127,  9s  to  156,  8s  to  69,  7s  to  32,  alternate  3s  and 
8s  to  14,  alternate  2s  and  4s  to  52.     (Limit  300) 

Grouping.  In  adding  columns  of  figures,  it  is  advisable  to 
group  digits  into  sub-totals,  and  to  combine  these  sub-totals. 
The  grouping  may  be  consecutive,  irregular  or  by  tens  and 
twenties. 


Illustrations. 

L  11.  III. 

3\  4\  5                 (I.)  Read  instantly  the  joined  numbers  as  single 

6  /  9  J  9  \          numbers.     The  consecutive  totals,  downward,  are : 

9\  3/  /4j        9,20,30,38,48. 

2  y  8  \  ^  ^  (I^*)  ^^^^  upward,  grouping  simple  numbers, 
4\  7\  ^6  even  if  non-consecutive.  Totals:  7,  16,  25,  33,  40,. 
6/  2)  8\        49. 

3  \  9  /  3  I  (III.)  Since  "tens"  are  easier  to  add  than  irreg- 
6/  Q\  I  2'  ular  combinations,  group  "tens"  as  shown. 
2\  1  /  \  7            Totals  upward:  5,  15,  25,  35,  45,  50. 

1  )  49  _5 

J^J  50 
48 


It  is  frequently  necessary  to  combine  groups  of  two-place 
numbers.  Some  accountants  regularly  add  double  columns 
in  place  of  single  columns.  Until  totals  are  possible  by  in- 
spection, add  the  tens  and  then  the  units,  thus:  46  +  57  = 
40  +  50  +  6  -t-  7. 


10  BUSINESS  ARITHMETIC. 

ORAL    EXERCISE. 

Name  the  sums  of  the  following: 

36     21     84     73     27     62     81     94  78  55  43  16  21  62  43  67 

43     69     72     69     41     25     73     18  63  57  54  11  21  29  13  58 

—    —     —     —     —     —     —     —  —  —  —  14  10  71  2  11 


Complements.  The  complement  of  a  number  is  the  dif- 
ference between  it  and  the  lowest  number  of  the  next  higher 
order.  Thus  the  complement  of  8  is  10  —  8,  or  2;  the  com- 
plement of  87  is  100  —  87,  or  13.  Complements  are  an  aid 
to  rapid  addition. 

Illustration.    376  +  96  =  376  +  100-4. 

ORAL    EXERCISE. 

1.  Name  the  complements  of  the  following:  7,  21,  36,  64,  77,  129,  973, 
9959. 

2.  When  is  it  advisable  to  use  the  complement  in  mental  addition? 

3.  Add  by  complements:  64  +  97,  88  +  93,  173  +  98,  77  +  84, 
568  +  92. 


EXERCISE. 

Add  these  colunms 

without  copying 

figures,  groupmg 

simple  numbe 

1. 

2. 

3. 

4. 

5. 

524 

3164 

629 

234,567 

1  234  567 

689 

43 

61324 

123,456 

2  456  673 

306 

721 

399 

912,345 

5  928  431 

1247 

2903 

2401 

891,234 

4  241  326 

1201 

8127 

88 

789,123 

5  261  252 

3824 

655 

1120 

678,912 

4  683  392 

6735 

432 

21205 

567,891 

9  221  243 

3289 

288 

4630 

456,789 

4  322  881 

6476 

179 

880 

345,678 

3  333  333 

6.  Why  is  column  1  easier  to  add  than  column  3? 

7.  What  aids  to  quick  addition  are  found  in  columns  4  and  5? 

8.  Write  two  original  columns,  one  of  20  six-place  figures,  the  other 
of  60  two-place  figures.     Add.  Which  example  is  more  difficult?     Why? 

9.  Name  ten  simple  applications  of  addition,  in  everyday  life. 


ADDITION.  11 

Checking.  All  addition  should  be  "checked,"  or  tested  by 
one  of  the  following  methods. 

(a)  Reverse  direction.  Add  down  after  adding  up,  etc. 
Why? 

(6)  After  adding  by  consecutive  numbers  or  groups,  group 
by  tens.     Why? 

(c)  Divide  the  column  at  one  or  more  points  by  horizontal 
lines.  Determine  the  total  of  each  partial  column.  Set  out 
these  totals  and  add. 

(d)  Add  the  unit  column,  writing  the  total  at  one  side; 
add  the  "tens"  column  independently,  offsetting  the  total 

as  shown.  Continue  the  process.  The 
total  of  the  sub-totals  should  equal  the 
column  total.  This  method  is  also  used  for 
direct  addition  by  one  who  is  likely  to  be 
interrupted  frequently.  One  may  stop  add- 
13168  13168  ing  at  any  point,  and  continue  again  when 
opportunity  offers. 

(e)  Casting  or  counting  out  nines. 

To  cast  out  nines  from  a  number,  add  its  digits  and  when- 
ever the  sum  equals  or  exceeds  9  discard  9,  adding  the  re- 
mainder to  the  succeeding  digits,  etc. 

To  check  addition,  discard  the  nines  from  the  numbers 
added  and  also  from  their  sum.  Discard  the  nines  from  the 
remainders  of  the  numbers  added.  If  the  final  remainder  equals 
the  remainder  of  the  sum,  the  sum  may  be  presumed  to  be 
correct. 


3216 

18 

2479 

15 

3821 

20 

3652 

11 

Illustration. 


Final  Remainder. 

426  3 

824  5 

936  0 

1542  3 


3728  2,  check  remainder. 


12 


BUSINESS   ARITHMETIC. 


WRITTEN    EXERCISE. 
Below  is  shown  a  portion  of  the  circulation  statement  of  a  newspaper, 
for  a  recent  year.    Find  and  check  the  monthly  totals,  using  the  methods 
just  outlined. 


Date 

January 

February 

March 

April 

May 

June 

1 

45578 

59471 

60105 

60150 

58408 

69636 

2 

505Jt2 

59723 

59886 

48302 

59392 

66880 

3 

56551 

59116 

60036 

59937 

59029 

56746 

4 

57318 

59400 

61096 

59887 

58713 

46006 

5 

57273 

49249 

49512 

61030 

58395 

57192 

7 

57942 

59522 

59469 

60105 

47000 

57745 

8 

46937 

59400 

60107 

6O4OO 

68410 

57679 

9 

57771 

59281 

60484 

47861 

58910 

67163 

10 

57909 

59613 

59960 

6O415 

686I4 

66662 

11 

58330 

59395 

59837 

60634 

58164 

46349 

12 

58533 

59035 

48821 

60485 

68084 

57165 

13 

58106 

59372 

601 49 

60332 

67966 

66571 

U 

58428 

59306 

59575 

59953 

47006 

57876 

15 

48O47 

59536 

59680 

59939 

67966 

57957 

16 

57797 

59469 

59839 

47505 

58102 

57859 

17 

57945 

59470 

60663 

58577 

57664 

66226 

18 

58443 

59477 

59950 

60825 

57335 

46547 

19 

58696 

49410 

48875 

60134 

57349 

57287 

20 

58393 

59526 

6OI43 

61332 

56774 

57684 

21 

58697 

59394 

60329 

60874 

46346 

57107 

22 

48676 

53016 

60260 

60504 

67296 

67208 

23 

58396 

60116 

59825 

48023 

57197 

56767 

H 

59334 

59823 

60366 

59989 

57927 

56350 

25 

59015 

59934 

60060 

69751 

67884 

46069 

26 

59453 

49187 

48110 

59658 

57362 

56293 

27 

59544 

59981 

60250 

59170 

57281 

56989 

28 

59075 

69732 

60023 

69147 

46522 

67203 

29 

49327 

60569 

68858 

57068 

66927 

30 

59580 

60316 

47412 

49684 

56919 

31 

59381 

60064 

57070 

JUiAfy 

ibia£ 

Horizontal  Addition.     It  is  often  necessary  to  find  the  sum 
of  several  numbers  written  in  a  horizontal  row,  without  re- 
writing them  in  vertical  columns.     Such  addition  is  common  in 
billing,  accounting  and  statistics. 
Illustration. 


2472 


3014 


5207 


Total  Col. 
13962 


ADDITION.  13 

In  adding,  follow  the  usual  rule.  Add  the  units  (22)  writing  the  unit  value 
of  the  total  and  carrying  '2';  add  the  tens,  etc.  Be  careful  of  positions 
after  passing  the  second  place. 

EXERCISE. 

Sales  recapitulation  sheets  are  used  in  department  stores,  factories, 
etc.,  to  sum  up,  for  jBxed  periods,  the  total  sales  of  each  department,  of 
each  class  of  output,  of  each  salesman,  etc.  Their  totaling  often  involves 
both  horizontal  and  vertical  addition,  as  in  the  following: 

Recapitulation  of  Department  Sales. 

February,  19 — . 

Day  Dept.  A  Dept.  B  Dept.  C  Dept.  D  Dept.  E  Dept.  F   Totals 

1  $460.77  $344.00  $450.55  $305.50  $355.06  $455.56     ? 

2  544.04   550.00   405.60   366.67   460.60   758.24 

3  477.07   637.89   400.33   536.56   367.89   457.65 

5  477.08  460.70  512.45  435.56  404.40  460.70 

6  546.87  315.85  428.67  345.67  546.82  467.19 

7  478.89  211.81  543.78  377.69  512.87  601.03 

8  213.56  110.87  232.16  215.46  127.83  105.68 

9  136.46  98.67  205.38  122.69  200.80  149.03 
10  568.67  511.84  690.57  477.89  624.35  589.98 

12  477.21  647.29  489.39  436.29  527.69  672.39 

13  567.85  437.27  512.11  532.56  455.21  532.24 

14  547.54  512.37  544.65  522.22  499.05  542.26 

15  600.25  590.01  504.71  488.34  473.56  532.56 

16  580.35  543.26  379.67  476.58  532.23  433.59 

17  555.88  496.36  665.00  299.99  548.84  477.25 

19  788.27  564.88  675.50  590.54  693.47  657.76 

20  433.19  459.23  435.67  467.89  590.12  534.56 

21  512.25  511.11  561.00  500.15  509.90  503.32 

22  485.20  500.44  503.67  483.74  468.49  589.70 

23  522.16  460.69  544.36  467.78  488.29  512.61 

24  501.44  577.10  600.00  503.92  605.58  505.45 

26  473.36   546.46   516.17   563.10   718.14   503.25 

27  494.58   536.25   500.26   468.87   600.21   580.41 

28  456.67   478.75   601.37   503.22   496.59   502.45 

Totals 

(a)  Total  the  vertical  columns.     What  do  they  show? 
(6)  Total  the  horizontal  columns.    WTiat  do  they  show? 
(c)  Suggest  a  "check"  for  the  vertical  and  horizontal  totals. 


14  BUSINESS  ARITHMETIC. 

(d)  Indicate  the  possible  effects  of  weather,  season,  special  sales,  and 
other  causes,  on  values  appearing  in  departmental  columns.  Show  what 
values  in  the  above  table  might  be  specially  accounted  for. 

Note.  Most  tabular  addition  in  business  offices,  and  considerable 
computation  requiring  other  fundamental  operations,  is  now  done  by 
machinery.  As  early  as  A.D.  968,  Gilbert,  Archibishop  of  Rheims,  per- 
fected some  ingenious  computing  machines,  which,  however,  sometimes 
failed  to  compute  correctly.  Pascal  (1642-45)  invented  a  machine  correct 
in  theory,  which,  while  not  practically  useful,  formed  the  basis  for  some 
modern  machines  of  great  capacity. 

The  modern  machines  generally  perform  the  fimdamental  operations. 
They  are  either  "listing"  or  "non-listing"  machines.  One  standard 
machine  has  eight  or  more  "banks"  of  keys,  each  containing  nine  digits 
and  having  its  own  place  value.  The  keys  to  represent  a  certain  number 
are  pressed  down  and  the  movement  of  a  lever,  or  the  pressing  of  an  electric 
button  causes  the  printing  of  the  number  on  a  continuous  roll  of  paper. 
The  next  number  is  printed  below  the  first,  the  changing  total  appearing 
in  the  machine.  A  special  movement  prints  the  final  total  on  the  paper, 
and  "clears  the  machine"  for  the  next  operation.  Some  machines  list 
abstract  numbers;  others  list  money  values,  fractions,  etc.  Some  are  at- 
tached to  typewriters  so  that  bills  of  merchandise  may  be  written  and 
automatically  totalled. 

TABULATIONS  AND  STATISTICS. 
EXERCISE. 

These  two  news  items  have  appeared  in  print: 

(Tabulated.)  (Untabulated.) 

The  Department  of  Agriculture  The  losses  caused  by  insects 

estimates    the    minimum    annual  to  the  products  of  the  country, 

damage  done  to  crops  by  our  most  during  an  average  year,  are  es- 

destructive  insects,  as  follows :  timated  by  the  Department  of 

Chinch-bug $60,000,000  Agriculture,  as  follows:  Cereals, 

Grasshopper 50,000,000  $200,000,000;    animal    products, 

Hessian  fly 40,000,000  $175,000,000;  forests  and  forest 

Corn-root  worm 20,000,000  products,  $111,000,000;  products 

Corn-ear  worm 20,000,000  in     storage,     $100,000,000;  hay 

Cotton-boll  weevil 20,000,000  and   forage,    $53,000,000;    truck 

Codling  moth  (Apple) .  20,000,000  crops,  $53,000,000;  cotton,  $50,- 

Armyworm 15,000,000  000,000;      tobacco,     $5,300,000; 

Cotton-boll  worm 12,000,000  fruits,       $27,000,000;       sugars, 

Grain  weevil 10,000,000  $5,000,000;  miscellaneous  crops, 

San  Jose  scale 10,000,000  $5,800,000.     Total,    | ? . 

Cotton-leaf  worm. ....     8,000,000 

Potato-bug 8,000,000 

Cabbage  worm 5,000,000 

Total ? 


ADDITION.  15 

(a)  Find  the  missing  totals.     In  which  case  is  it  easier?     Why? 

(6)  Wliich  statement  is  read  more  easily?     Which  is  more  noticeable? 

(c)  From  which  statement  is  it  easier  to  select,  or  reclassify  values? 
Why? 

(d)  Compare  capitahzation  in  the  two  cases.     Compare  use  of  "$." 

(e)  Tabulate  the  second  illustration,  using  a  brief  title  in  place  of  the 
introductory  clause. 

The  illustrations  given  above  show  the  value  to  the  statis- 
tician of  the  "column"  and  "total"  in  conveying  numerical 
information.  The  column  tabulation  attracts  attention,  and 
brings  each  numerical  value  in  sharp  contrast  with  other 
values,  while  allowing  comparison  with  the  total.  In  tabulat- 
ing, names  attached  to  values  should  be  brief,  but  definite. 
Each  tabulation  should  have  a  brief  descriptive  title. 

EXERCISE. 

1.  Tabulate  and  find  the  lowest  cost  of  construction  of  a  building  for 
which  the  following  bids  were  submitted: 

For  masonry,  iron  work,  carpentry,  glazing,  tinning,  plastering,  etc. — 
A.  B.  Ryan,  $49,845;  J.  D.  Norton,  $41,900;  C.  A.  Olhaus,  $42,344;  Robert 
Adams,  $48,275;  Brown  &  Mann,  $35,691;  Wm.  Harris,  $37,573;  W.  B. 
Norton,  $44,678;  A.  S.  Keene,  $43,536;  Northern  Cons.  Co.,  $40,547;  Jas. 
Davis,  $35,386. 

For  materials  and  labor  necessary  to  furnish,  install  and  complete  a 
low  pressure,  steam-heating  apparatus: — Robert  Davis,  $4,646;  Morris 
Heating  Cons.  Co.,  $3,674;  Evanson  &  Son,  $3,940;  E,  T.  Roberts  Co., 
$3,970;  Barton  Mfg.  Co.,  $4,180;  Geo.  A.  Sanderson,  $4,900;  Brown  &  Mann, 
$4,950;  Wm.  Harris,  $3,398. 

For  materials,  labor,  etc.,  for  plumbing  work: — D.  F.  Kean,  $3,325; 
Robert  Harris  Co.,  $5,132;  Wm.  Harris,  $4,736;  A.  B.  Ryan,  $4,728;  Harper 
&  Son,  $4,198;  Brown  &  Mann,  $5,763. 

From  the  tabulation  just  made,  find :  (a)  what  bidder  will  complete 
the  work  for  the  lowest  sum;  (h)  the  lowest  sum  for  which  the  work  may 
be  done  by  giving  sub-contracts  to  different  bidders. 

Decimal  Addition. 

INTRODUCTORY   EXERCISE. 
1.    Annex  a  cipher  to  .6,  and  read  result. 


16  BUSINESS  ARITHMETIC. 

2.  Reduce  4  to  tenths. 

3.  Reduce  53  to  hundredths, 

4.  Add  .3  and  .6. 
6.  Add  A,  .6,  .9,  .8. 

6.  Add  .04,  .83,  .07,  001. 

7.  Add  .047  and  .07.     (.07  =  how  many  thousandths?) 

8.  How  does  reduction  to  a  common  order  aid  in  decimal  addition? 

There  is  practically  no  difference  between  integral  and 

decimal  addition.     Integral  or  decimal  units  of  the  same  order 

are  added.     In  writing  in  column,  be  sure  that  the  decimal 

points  are  in  vertical  alignment,  as  this  will  ensure  proper 

"order"  position,  and  also  the  position  of  the  decimal  point 

in  the  total. 

Illustration. 

46.205 

392.0872 

1245.01232 

254.67005 


1937.97457 


ORAL   EXERCISE. 

1.  How  does  the  number  of  decimal  places  in  the  total  compare  with 
the  nimiber  of  places  in  the  constituent  numbers? 

2.  May  the  total  contain  a  less  number  of  decimal  places  than  any 
significant  number? 

Add: 

3.  .4  and  .85.  8.    2.33,  1.062  and  2.0382. 

4.  .072  and  0096.  9.  3.25,  1.0006  and  8.0129. 
6.  .12,  .065  and  .363.            10.  .46,  .05,  .8  and  .002. 

6.  .013,  .21,  and  .220.  11.  6.3205  and  .00061. 

7.  .01,  2.006  and  15.0126.         12.  .001,  2  01  and  .0352. 


ADDITION. 

EXERCISE. 

Total  these  columns: 

(1) 

(2) 

(3) 

23.075 

123.456729 

4.37 

06.5082 

54.08296 

83.070096 

16.09327 

5.4729825 

5.9276 

5463.382473 

.007369 

329.832 

1.29634 

527.8308896 

4.00892 

50069.8002173 

5.2100009 

.0006732 

542.06345276 

67.31438 

1.01 

12.47234 

5729.61 

15.2729 

5067.30821 

80.8 

.888005 

Note.    A  large  proportion  of  decimal  addition  deals  with  mone- 

(4) 

(5) 

(6) 

$  326.925 

£  456.29 

45.B25  fr. 

48.065 

39.625 

129.8 

723.8 

1233.40 

46239.03 

9654.47 

56.1235 

596.008 

830.425 

5.60 

63.9 

65.72 

1283.55 

1321.48 

12756.23 

14756.125 

32635.835 

52.08 

677.0875 

4.006 

926.00 

1239.05 

5282.56 

7259.12 

36.075 

43.45 

5.69 

5296.5125 

6.735 

627.33 

83.0125 

529.62 

46.09 

962.825 

88.9 

17 


7.  Find  the  totals,  in  units  of  a  thousand  feet,  of  these  orders  for 
lumber:  6246  ft.,  15,389  ft.,  7500  ft.,  4382  ft.,  965  ft.,  45  ft.,  12,925  ft.,  6750  ft. 

8.  Total  these  engineering  measurements,  and  express  total  in  units 
of  1000  ft.:  564',  296.5',  4521',  885.2',  750',  1325',  632.8',  983.7',  66.8'o 


CHAPTER  IV. 
SUBTRACTION. 

INTRODUCTORY    EXERCISE. 

1.  Name  the  difference  between  2  and  9,  16  and  21, 

2.  From  129  subtract  4s. 

3.  From  138  subtract  alternate  3s  and  5s. 

4.  Give  several  illustrations  of  the  use  of  subtraction  in  everyday  life. 

Subtraction  is  one  phase  of  addition.  It  is  the  process  of 
finding  the  number  to  be  added  to  the  lesser  of  two  given 
numbers  in  order  that  the  sum  may  equal  the  greater. 

lLL'j3rRATiON.     Find  the  difference  between  774  and  1969. 

Process.     Add  to  the  subtrahend,  digit  by  digit,  until 
iQftQ  miniioTiri  *^®    miuucud    is    obtained,   thus:    4    (sub.)  +5  =  9 

774  SSend      ("^^^•)-     '^"^^  ^'  ^«  "^^^  difference.     7  (sub.)  +  9  =  16 
n^  suDtranena.     ^^^^^^    ^^^^^  9  ^^  ^^^,^  pj^^^  ^^  difference.     7  (sub.) 

1195  difference.        +  1  =  8  (9  -  1,  min.).    0  (sub.)  +  1  =  1  (min.). 

Note.  In  formal  work,  it  is  often  inconvenient  to  write  the  lesser 
number  under  the  greater.  Position,  however,  makes  no  difference  in 
method.    The  above  example  might  be  written: 

(a)      774  sub.  (b)     1969  -  774  =  1195. 

1969  min. 
1195  diff. 

EXERCISE. 

1.  Show  that  these  subtraction  "checks"  are  true. 

(a)    The  sum  of  subtrahend  and  difference  equals  minuend. 
(6)     The  minuend  less  the  difference  equals  the  subtrahend. 
Solve  by  inspection: 

2.  Find  by  addition  the  complements  of  36,  29,  64,  83,  91,  896. 

3.  Solve  by  addition: 

54  -  13      76  -  23      148  -  93       168  -  72 

59  -  37      43  -  29       124  -  58      249  -  68 

18 


SUBTRACTION.  19 

Note.  Oral  work  is  sometimes  simplified  by  subtracting  the  orders 
of  the  subtrahend  independently,  beginning  with  the  highest.  This  is 
termed  subtraction  from  the  left.  Thus  578  -  364  =  578  -  300  (278) 
-  60  (218)  -  4  =  214. 

4.  Solve,  by  subtraction  from  the  left : 

249  -  67  328  -  243  1527  -  892  968  -  871 

1116  -  452  1793  -  896  529  -  381  4473  -  269 

Note.  Irregular  numbers  may  be  subtracted  mentally  by  subtracting 
an  even  number  of  the  next  higher  order,  correcting  the  difference  of  the 
numbers  by  adding  the  difference  of  the  subtrahends.  Thus  384  —  47 
=  384  -  50  (334)  +  3  =  337. 

5.  Solve,  explaining  the  process: 

362  -  98  473  -  57  734  -  96  263  -  99 

654  -  319  328  -  246  899  -  473  346  -  228 

Note.  Often,  in  practical  work,  the  subtrahend  must  first  be  obtained 
by  addition. 

6.  624  -  (32  +  16  +  11)  =  ? 

7.  1754  -  (200  +  311  +  40)  =  ? 

8.  Sales  of  72,  14  and  11  pounds  from  a  stock  of  360  lb.  leave  how 
many  pounds  on  hand? 

EXERCISE. 

1.  59,768,003  -  407,019  =  ? 

2.  $3,876,547  3.    $567,839  4.    987,654,321 

108,976  203,678  123,456,789 


?  ?  .  ? 

In  the  following,  subtract  the  last  number  from  the  sum  of  the  balance 
of  the  colunm: 


7239 

6.  1209 

7. 

213 

8.  32139 

2146 

9660 

32 

4781 

1299 

5138 

921 

61020 

881 

1234 

4238 

3121 

4731 

8108 

1119 

5288 

9-12.     Repeat  the  last  four  examples,  subtracting  the  first  number 
from  the  sum  of  the  balance  of  the  colunm. 

Note.     Subtraction  is  often  combined  with  direct  addition,  especially 
in  business  forms  and  statistical  tables. 

13.     This  form  is  one  of  many  used  in  business  to  keep  account  of  stock 
on  hand. 


20 


BUSINESS   ARITHMETIC. 


STOCK  RECORD  OF 


LIMIT 

MO 


WHERE  KEPT 


STOCK  RECEIVED 

STOCK  SOLD 

ON 
HAND 

DATE 

FROM  WHOM 

QUAN. 

DATE 

TO  WHOM 

QUAN. 

QUAN. 

19— 

19- 

3-3 

^rtAP&nid^'u, 

172 

3-5 

*^3762 

9 

163 

3-8 

3964 

27 

3-9 

4287 

15 

3-12 

4561 

42 

3-21 

ja/rtAoifn,    Vo. 

50 

3-21 

5265 

18 

3-24 

5290 

54 

3-26 

5380 

21 

3-29 

5620 

20 

(a)  Extend  the  form. 

(6)  Explain  the  writing  of  the  date. 

(c)  How  are  entries  in  the  "On  hand"  column  obtained? 

(d)  Explain  the  solution  for  Mar.  21. 

(e)  How  many  desks  are  on  hand  on  Mar.  11? 

(/)  How  may  the  "Received"  and  "Sold"  columns  be  checked? 

14.  Many  private  families,  living  on  a  "cash"  basis,  keep  simple 
record,  showing  in  detail  household  income  and  expenditures.     (Page  21.) 

(a)  Explain  the  first  items  in  the  "Total  paid  out"  and  "Balance" 
columns. 

(6)     Find  the  remaining  values  for  these  columns. 

(c)  Total  all  the  vertical  columns  except  the  "Balance"  column. 

(d)  Why  is  not  the  "Balance"  column  totaled? 

(e)  Find  a  check  for  the  "Total  paid  out"  total. 
(/)     Suggest  a  check  for  the  final  balance. 

(g)  Suggest  how  amounts  in  the  different  columns  might  vary  with 
the  season  of  the  year. 

The  statistician  who  is  tabulating  crops,  wealth,  railway 
mileage,  bank  clearings,  exports,  etc.,  frequently  gives  figures 
for  earlier  periods  and  shows  by  a  difference  column  the  amount 
of  change,  the  differences  often  lending  special  significance  to 
the  gross  values.     The  business  man,  also,  gives  special  atten- 


*  Order  number  identifies  purchaser. 


SUBTRACTION. 


21 


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22  BUSINESS  ARITHMETIC. 

tion  to  the  increases  or  decreases,  as  well  as  to  the  total  of  his 
sales,  profits  or  expenses.  In  the  exercise  that  follows,  note 
how  the  differences  add  weight  to  the  presentation  of  facts. 

EXERCISE. 

1.  These  figures  from  a  financial  article  show  certain  facts  concerning 
the  business  of  the  country  for  two  consecutive  years.  Fmd  the  differences, 
placing  decreases  in  the  "increase"  column,  but  marking  them  with  a 
minus  sign. 

Two  Years'  Record. 

1910  1911  Increase 

Bank  clearings $158,450,000,000  $162,914,100,000             ? 

Railway  earnings 2,395,800,000  2,423,400,000             ? 

Exports 1,867,605,000  1,637,256,000             ? 

Imports 1,392,500,000  1,426,200,000             ? 

Grain  crops 2,702,458,000  2,489,000,000             ? 

EXERCISE. 

1.  Bring  to  class  four  illustrations  of  "  difference  "  columns  in  statistical 
tables. 

2.  Sum  up  the  different  methods  of  checking  addition  and  subtraction 
used  or  suggested  in  this  chapter. 

Subtraction  of  Decimals. 

INTRODUCTORY    EXERCISE. 

1.  From  36  hundredths  subtract  12  hundredths. 

2.  5.46  -  2.39  =  ? 

3.  .62  =   ?  thousandths.     .52  -  .412  =  ? 

The  only  difference  in  process  between  whole  nun^iber  and 
decimal  subtraction  consists  in  the  necessity  for  a  reduction, 
in  the  latter,  of  both  subtrahend  and  minuend  to  decimals  of 
the  same  order. 

Illustration.    Find  the  difference  between  5.029  and  3.0014. 
Solution.    5.029  is  the  same  as  5.0290.        5.0290 

3.0014 


2.0276 


SUBTRACTION.  23 

EXERCISE. 
By  inspection  find  the  difference  between: 

1.  .51  and  .402.  6.     .01024  and  .08. 

2.  3.6  and  1.32.  7.     .000004  and  .0004. 

3.  .9  and  .09.  8.     10.003  and  .230. 

4.  1.  and  .0321.  9.    From  3.75  subtract  .05s. 

5.  2.0004  and  1.3.  .  10.     From  .324  subtract  .004s. 
11.    From  one  and  one-hundredth,  subtract  nine  ten-thousandths. 

EXERCISE. 

1.  4.0739625  -  2.00100376  =  ? 

2.  5506.38  -  296.54325  =  ? 

3.  100,109.0001003  -  1202.12345  =  ? 

4.  From  46  subtract  the  first  of  the  following  decimals;  from  the 
remainder  subtract  the  second,  and  so  on:  5.0621,  2.010309,  1.0042,  .98396, 
25. 1013096, 1 .24965.     How  may  the  accuracy  of  the  final  answer  be  checked? 

Decimal  subtraction  is  common  in  higher  mathematics  and 
in  scientific  and  estimating  problems.  Two-place  decimal 
subtraction  is  common  in  money  computations.  Usually  sub- 
traction is  incidental  to  some  other  process. 

EXERCISE. 

1.  A  farmer's  poultry  account  for  January  is  as  follows:  Value  of  fowls, 
Jan.  1,  $1800;  sales  of  fowls  during  the  month,  $247.53;  value  of  fowls 
at  close,  $1750;  cost  of  food  and  care,  $112.50;  sales  of  eggs,  $105.82; 
miscellaneous  expenses,  $54.27.  Make  a  neat  statement  showing  the  net 
profit  for  the  month. 

2.  A  metal  bar  16  inches  long,  is  contracted  under  pressure  .8583765 
inches.     What  is  its  length  imder  pressure? 

3.  A  yard  measure  is  equal  to  0.914,402  meters.  A  meter  exceeds  a 
yard  by  what  decimal  of  itself? 

4.  A  certain  cloth  shrinks  .1525  of  a  foot  per  yard  in  dyeing.  A  yard 
of  the  undyed  cloth  produces  how  many  yards  of  dyed  material? 

5.  The  needle  on  a  recording  machine  travels  from  3.256  ft.  on  the 
scale  to  4.105  ft.,  a  distance  of  ?  ft. 


24 


BUSINESS  ARITHMETIC. 


6.  In  one  city,  having  5c  car  fares,  the  cost  of  carrying  each  passenger 
is  3.9246c.  In  another  city,  having  a  4c  fare,  the  cost  is  3.1024c.  Find 
the  profit  per  passenger  in  each  case,  and  the  difference  in  cost  per  city. 

7.  Complete  this  memorandum  of  costs  and  profits. 


The  Novelty  Manufacturing  Company. 
Memorandum  of  Costs  and  Profits  per  Article. 


Article. 

Cost  to  Make,  ^. 

Net  Sell.  Pr.  ff. 

Net  Profit. 

Butterfly  cards 

Dime  savings  banks 

Remembrance  cards     ... 

3.0245 
14.308 
.0965 

4.85 
13.0625 

7.5 
16.5 

2.15 

6.1125 
15. 

? 
? 
? 

Surprise  box             

?" 

Three  of  a  kind 

? 

CHAPTER  V. 
MULTIPLICATION. 

Multiplication,  as  a  number  process,  is  second  in  importance 
to  addition,  and  has  extensive  and  varied  applications.  It  is 
really  an  addition  "short  cut,"  for  expressions  of  multiplica- 
tion, such  as  "5  X  381"  may  be  written  in  the  form  "381 
+  381  +  381  +  381  +  381."  As  an  independent  process, 
also,  it  lends  itself  to  so  many  short  methods  as  to  make  it 
worth  a  special  study. 

Illustration.    Multiply  3264  by  529. 

3264  (multiplic^d) 
529  (multiplier) 

29376  (units) 
6528    (tens) 
16320      (hundreds) 


1726656  (product) 
In  the  product  there  are  9  X  3264  units;  2  X  3264  tens  and  5  X  3264 
hundreds. 

Note.    The  multiplicand  and  multiplier  are  called  factors  of  the  product. 

While  the  multiplier  is  invariably  an  abstract  number,  the 
multiplicand  and  product  may  be  either  abstract  or  concrete, 
but  they  are  always  like  numbers.  As  a  result,  the  product  is 
not  altered  by  changing  the  order  of  the  factors  considered  as 
abstract  numbers. 

Illustration.  The  weight  of  8724  bx.  of  65  lb.  each  is  either  (8724X65) 
lb.,  or  (65  X  8724)  lb. 

FOR    DISCUSSION. 

1.  What  is  the  objection  to  solving  the  examples  illustrated 
above  by  addition? 

2.  At  what  point,  in  regular  multiplication,  does  addition 
occur? 

25 


26 


BUSINESS   ARITHMETIC. 


3.  In  finding  the  product  of  873,926  by  1001,  which  factor 
should  be  used  as  multiplier?     Why? 

4.  Why  is  this  statement  incorrect: 

"$16  X  15  bbl.  =  $180"? 

ORAL    EXERCISE. 

1.  Find  the  products  of:  3  X  14,  7  X  18,  3X3X6,  4  X  2  X  22, 
8  X  40  bu  ,  5  X  81c,  2  X  3  X  8  X  4,  2  X  24  ft. 

2.  Multiply  each  integer  under  21,  by  itself  and  by  each  smaller 
integral  number.     These  are  the  most  used  factors. 

3.  A  camping  party  of  three  men,  planning  a  sixty  day  trip,  order 
rations  in  the  proportions  given  below.  Determine  the  required  quantity 
of  each  article. 

Supplies  per  Man  per  Thirty  Days. 


Flour 241b. 

Oatmeal 7  " 

Beans 6   " 

Sugar 12   " 

Cond.  milk.  ...   3  en. 


Tea,  cocoa..  ,1  lb. 

Coffee 2  " 

Rice 4  " 

Salt 1   " 

Prunes 2  " 


Butter....  21b. 
Bak.  pow..  1  " 
Bacon.... 11  " 
Pepper ...    1  "   (6  men) 


4.  A  certain  contractor  foimd  constant  need  for  the  products  of  the 
factors  80  to  100  by  the  factors  20  to  40.  He  prepared  the  form  outlined 
below,  the  product  of  each  two  factors  being  written  at  the  intersection 
of  the  proper  horizontal  and  vertical  columns.  The  product  of  84  by  22 
is  shown.  Compute  the  other  products  mentally.  Show  how  to  use  an 
addition  method  for  the  work.  Suggest  a  method  of  checking  the  compu- 
tations. 


80 

81 

82 

83 

84 

85 

Etc. 

20 

21 

22 

1848 

Etc. 

5.     Construct  a  similar  multiplication  table  for  the  factors  1  to  25  in- 
clusive. 

EXERCISE. 

Find  the  products  of  the  following  and  check  accuracy  by  reversing 
factors: 


MULTIPIJCATION.  ,  27 

1.  726,943  X  120,301.  4.  373,259  X  129. 

2.  3,629,007  X  20,905.  5.  12,345  X  67,890. 

3.  16,730,021  X  802,001.  6.    20,103  X  1,387,654. 

7.  A  railway  company  sends  out  eight  engineering  parties  of  19  men 
each,  on  reconnoisance  and  track  location  work.  Rations  are  required 
for  200  days,  based  on  a  U.  S.  Government  ration  Ust.  Prepare  a  tabu- 
lation showing  the  quantity  of  suppUes  to  be  provided. 

Ration  List  for  One  Man  for  One  Hundred  Days. 
(Take  first  named  article.) 
100  lb.   fresh  meat,  including  fish  and  poultry. 
50  "    cured  meat,  canned  meat,  or  cheese 
15   "    lard. 

80  "    flour,  bread,  or  crackers. 

15  "    corn  meal,  cereals,  macaroni,  sage,  or  com  starch. 
5  "    baking  powder,  or  yeast  cakes. 
40  "    sugar. 

1  gal.  molasses. 
121b.   coffee. 

2  "    tea  or  cocoa. 

10  en.  condensed  milk,  or  50  qt.  fresh  milk. 
101b.  butter. 

20  "    dried  fruit,  or  100  lb.  fresh  fruit. 
20  "    rice  or  beans. 

100  "    potatoes  or  other  fresh  vegetables. 
30  "    vegetables  or  fruits. 

4  oz.  spices. 

4  "    flavoring  extracts. 

8  "    pepper,  or  mustard. 

3  qt.  pickles. 
1  "  vinegar. 
41b.   salt. 

8.  As  a  first  result  of  the  survey,  the  Company  decides  to  build  19 
miles  of  single  and  38  miles  of  double  track.  Compute  the  total  quantity 
of  the  following  supplies  which  are  to  be  used  in  the  construction:  Cross- 
ties,  at  rate  of  2640  per  mile  of  single  track;  steel  rail,  at  90  pounds 
per  yard;  spikes,  at  the  rate  of  11,380  per  mile  of  single  track;  68  angle 
irons  for  special  construction,  34  feet  long,  weighing  16  poimds  per  foot; 
240  I-beams,  28  feet  long,  weighing  65  pounds  per  foot.  Give  quantities 
of  iron  and  steel  by  weight. 


28  BUSINESS  ARITHMETIC. 

Short  Methods. 
Most  fundamental  processes  of  number  may  be  abbreviated 
— often  with  marked  increase  in  rapidity  of  solution  and 
accuracy  of  result.  The  short  methods  used,  however,  should 
be  such  as  one  naturally  develops  for  himself,  or  those  that 
instinctively  appeal  to  one  as  being  simpler  and  more  direct 
than  the  regular  methods.  Memorized  rules  for  short  methods 
are  dangerous  aids,  as  an  inexact  memory  may  cause  an  in- 
correct application  of  the  rule  at  some  critical  moment.  Short 
methods  are  commonly  used  to  do  away  with  computation 
on  paper,  or  to  simplify  such  computation,  or  to  check  the 
accuracy  of  computations  performed  by  other  methods. 

Short  Methods  in  Multiplication. 

Some  typical  cases  follow.  Suggest  other  methods.  In 
each  case,  show  wherein  the  value  of  the  method  consists. 

Multiplying  by  10,  100,  etc.     726  X  10  =  ;  1428  X  100 

= ;  345  X  1000  = .     Compare  the  significant  figures 

of  the  product  with  those  of  the  multiplicand.  How  may  the 
product  of  such  factors  be  written  at  sight?  What  effect 
has  the  annexing  of  a  cipher?     Of  two  ciphers? 

ORAL   EXERCISE. 

1.  167X10  =  ?  4.     100X26  lb.  =  ?         7.     1,000X80,201=? 

2.  100X3,802  =  ?  5.     10,000X131  ft.  =  ?  8.    70,073X100,000  =  ? 

3.  100X50,201=?         6.     10X7,384  yd.  =  ? 
Find  the  weight  of: 

9.    246  bbl,  averaging  100  lb.  ea. 
10.     1,246  loads,  averaging  1000  lb. 

Multiplying  by  11,  21,  SI,  etc.,  and  by  multiples  of  11,  Outline 
a  short  method  for  these  multipliers,  suggested  by  the  follow- 
ing illustrations: 

(1)  11  X  64  =  10  X  64  +  64  =  640  +  64  =  704. 

(2)  31  X  126  =  3  (10  X  126)  +  126  =  3780  +  126  =  3906. 

(3)  101  X  48  =  100  X  48  +  48  =  4800  +  48  =  4848. 


MULTIPLICATION.  29 

ORAL    EXERCISE. 
Find  the  product  of: 

1.  11  X  16.        3.     11  X  242.        5.    21  X  360.  7.    21  X  19. 

2.  21  X  28.        4.    31  X  18.  6.    41  X  22.  8.     1001  X  287. 
Find  the  weight  of:                          Find  the  cost  of: 

9.     240  ft.  iron  at  111b.  per.  ft.      11.     12  lb.  butter,  @  31c. 
10.     165  kegs,  31  lb.  each.  12.     142  sq.  yd.  tiling,  @  21c. 

Multiplying  by  9, 19, 29,  etc.     After  studying  the  illustrations, 
outline  the  short  method  and  compare  with  the  previous  case. 

Illustrations. 

(1)  9  X  148  =  10  X  148  -  148  =  1480  -  148  =  1332. 

(2)  19  X  320  =  2  (10  X  320)  -  320  =  6400  -  320  =  6080. 
Used  in  paper  work,  also. 

(3)  Example.     Multiply  32,689  by  19. 

Long  method.  Short  method. 

32689  Factors  not  re-written. 

19  653780  (2  X 10  X  32689,  written  at  sight) 


294201  32689(1X32689) 

32689  621091  (by  subtraction) 

621091 

Note.     The  short  method  substituted  the  multiplication  by  2  for  the 
more  difficult  multiplication  by  9. 

ORAL    EXERCISE. 
Find  the  product  of: 

1.  9  X  87.  3.    29  X  16.  5.    9  X  269.  7.    29  X  17. 

2.  19  X  42.  4.     109  X  120.  6.     9  X  424.  8.     59  X  11. 
9.    Find  the  weight  of  720  pc.  castings,  weighing  9  lb.  each. 

EXERCISE. 

Using  short  methods,  find  the  product  of: 

1.  19  X  32,967.  4.    29  X  6,376,951.  7.    99  X  8,703,129. 

2.  9  X  1,213,027.  5.     19  X  12,587,328.         8.     1,009  X  784,562. 

3.  39  X  47,201.  6.     109  X  630,851. 

Multiplying  hy  5,  15,  etc. 

Illustrations. 

(1)  5  X  620  =  One-half  of  10  X  620  =  ^^  =  3100. 

(2)  15  X  48  =  10  X  48  +  H  of  10  X  48  =  480  +  240  =  720. 


30  BUSINESS   ARITHMETIC. 

ORAL    EXERCISE. 

1.  In  multiplying  a  number,  such  as  7,268,316,  by  5,  why  may  it  b« 
simpler  to  use  the  short  method? 

2.  Multiply  by  5:  16,  21,  32,  54,  123,  1309,  677,  1893. 

3.  Multiply  by  15:  24,  68,  152,  19,  137,  240,  62,  192,  1012. 

4.  Find  the  costs  of:  124  en.  at  5c;  16  lb.  nails  at  5c;  68  yd.  dress 
goods  at  15c. 

EXERCISE. 

Using  short  methods,  and  omitting  the  re-writing  of  factors,  find  the 
products  of: 

1.  5  X  7236.  3.     15  X  724  yd.  5.     15  X  8039  qt. 

2.  15  X  83,963  lb.         4.     5  X  12,463  qt.  6.     15  X  7204  oz. 

Multiplying  by  factors  of  multiplier. 

Illustration  :  36  X 120  =  6  X 6  X 120  =  6  (6  X 120)  =  6 X720  =  4320. 

ORAL    EXERCISE. 
i.     Compare  the  short  and  the  direct  methods. 

2.  Is  the  short  method  of  real  advantage  in  oral  or  in  written  work? 

3.  How  do  the  factors  themselves  limit  the  use  of  the  method? 

4.  Reduce  to  two  or  mor€  simple  factors:  65,  24,  49,  18,  72,  96,  48, 
45,  54,  84,  110,  39,  92,  120. 

5.  Determine  these  products  by  factoring  multiplier,  and  check  by 
other  short  methods: 

25  X  160  122  X  12  99  X  48  16  X  32 

240  X  18  12  X  118  15  X  72  20  X  155 

'     Multiplying  by  parts  (multiplication  from  the  left). 

Illustration.     (1)  Find  the  weight  of  722  bbl.  of  112  lb.  each. 
112  X  722  =  100  X  722  + 10  X  722  +  2  X  722  =  72200+7220+1444  =  80864 
(2)  Written  work.     Multiply  738,421  by  321. 

Regular  method.  Method  by  parts. 

738,421  Factors  not  re-written. 

321  221,526,300  (3  X  100  X  number) 

738421  14,768,420  (2  X  10  X  number) 

1476842  738,421  (1  X  number) 

2215263  237,033,141 

237033141 


MULTIPLICATION.  31 

Note.  The  advantage  of  this  method  for  oral  work,  lies  in  adding  con- 
tinuously decreasing  sub-totals  to  a  growing  total,  each  new  product  being 
added  as  found. 

ORAL  EXERCISE. 
Find  the  product  of: 

1.  115X36.         3.     23X42.  5.     62X312.  7.     103X428. 

2.  111X249.       4.     22X124.  6.     210X122.        8.     51X122. 

9.  What  is  the  output  in  16  hours  of  a  machine  making  240  doz.  cans 
per  hour? 

EXERCISE. 
Find  the  product  of: 

1.    212X324,567.        2.     1325X88,459,876.  3.     1245X87,960,345. 

SPECIAL  EXERCISE. 

1.  Write  a  brief  illustrative  paper  on  "Short  methods  in  Multipli- 
cation." 

2.  Write  a  paragraph  on  the  dangers  and  sources  of  error  in  the  use 
of  short  methods. 

3.  ^rite  a  brief  paper  on  "Methods  of  Checking  Multiplication." 

DECIMAL   MULTIPLICATION. 
INTRODUCTORY    EXERCISE. 

1.  One-tenth  of  6  =  ?  One-hundredth  of  6  =  ? 

2.  Ten  times  .6  =  ?  One  hundred  times  .6  =  ? 

3.  Compare  these  numbers  as  to  value:  .06,  .6,  6,  6000. 

4.  How  is  the  value  of  a  significant  figure  affected  by  moving  the  decimal 
point  to  the  right. 

5.  Suggest  a  short  method  for  multiplication  of  decimals  by  10,  100, 
1000,  etc.     Illustrate,  using  .006  and  4.2  as  multiplicands. 

6.  One-tenth  of  68  =  68  ^?;     or  68  X  ? 

7.  Using  72.6  show  the  effect  on  its  value  of  moving  the  decimal  point 
to  the  left. 

8.  Suggest  short  methods  of  multiplication  by  .1,  01,   001,  etc. 

The  process  of  multiplication  of  decimal  fractions  differs 
in  no  essential  from  the  midtiplication  of  integers,  except  in 
the  determination  of  the  decimal  point  of  the  product.  Notic- 
ing  that    .1  X  .01  =  1/10  X  1/100  =  1/1000    or    .001,    it   is 


32 


BUSINESS   ARITHMETIC. 


evident  that  the  number  of  decimal  places  in  the  product 
equals  the  sum  of  those  in  the  factors.  In  multiplication, 
therefore,  decimals  may  be  treated  as  integers,  the  proper 
number  of  places  being  pointed  off  after  the  numerical  product 
is  obtained. 


Illustrations.     (1)  Multiply  5268.39  by  1.6. 

Solution  and  analysis.    5268.39  contains  526,839  hundredths. 

1.6  contains  16  tenths 
526,839  hundredths  X  16  tenths    =  8,429,424  thousandths, 
or  8429.424 

(2)  Find  the  product  of  7.26  multiplied  by  3600. 

Solution.     3600  X  7.26  =  36  X  100  X  7.26  =  36  X  726. 

Note.     Notice  how  the  decimal  is  cancelled  before  the  multi- 
plication of  significant  figures  is  begun.     This  is  often  possible. 


5268.39 
1.6 

3161034 
526839 
8429.424 

726 

36 

4356 

2178^ 

26136 


(e)  .00008X4.00007 
(/)  9.001X1.0000401 

(g)   1.002  X. 021 
(h)  .06X6000 
(i)  7.25 X. 0002 


ORAL    EXERCISE. 

1.  How  many  decimal  places  in  the  products  of: 
(a)  3.6 X. 004  (c)  58 X. 020 
(6)  50.29 X. 00609      (d)  1573.829 X .032 

2.  Find  the  products  of: 
(o)  1.2X.12  (d)  4003X.0002 
(6)  .003 X. 0004  (e)  6.01  X. 005 
(c)  2.01  X. 00002         (/)  .000003 X. 3 

3.  Compute  the  cost  of: 

(a)  700  lb.  @  32c.     (b)  2000  bu.  @  45c.        (c)  2960  lb.  @  Jl.lO. 

EXERCISE. 
(Check  all  products.) 

1.  Multiply  7206.021  by  3.001029. 

2.  Multiply  739.65482  by  549.03296. 

3.  Multiply  7.2098  by  .02604,  and  the  product  by  100.09. 

ORAL    EXERCISE. 
*      (Use  the  short  methods  given  for  integral  numbers.) 

1.  100X26yd.  =  ?  4.     1000 X. 005  yd.  =  ?  7.     9X.8  =  ? 

2.  380X.2X.01  =  ?         5.    21X.4  =  ?  8.     19Xl.2  =  T 

3.  11X3.6  =  ?  6.     101 X. 08  =  ? 


MULTIPLICATION.  33 


GENERAL    EXERCISE. 

Note.  In  the  computation  of  costs  and  quantities  by  multiplication, 
decimals  are  usually  involved.     The  following  are  tj'pical  illustrations. 

1.  For  accounting  purposes,  Martin  &  Co.  take  an  inventory,  or 
valuation  of  their  stock  of  skins,  pricing  them  at  the  lowest  market  quo- 
tation, except  for  the  gray  fox  which  they  price  at  the  highest  rate.  Using 
the  following  quotations,  make  a  neat  tabulation  showing  the  valuation  of 
the  entire  stock. 

WOOL  AND  HIDES.— Quotations  for  fm-s  on  No.  1  articles  only. 
Wool,  washed,  free  of  burrs,  per  lb.,  38a40;  wool  unwashed,  per  lb.,  30a33; 
hides,  green,  per  lb.,  10;  dry,  per  lb.,  15al7;  sheepskins,  green,  each,  1.00a 
1.25;  dry,  each,  25a75;  calf -skins,  green,  each,  1.00al.50;  muskrat,  each, 
12al8;  skunk,  each  25a75;  mink,  each,  3.00a4.00;  rabbit  skins,  each,  10; 
opossum,  each,  25a27;  raccoon,  each,  25a90;  fox,  red,  each,  2.00a2.50; 
gray,  each,  75a90. 

The  stock  of  Martin  &  Co.  consists  of:  2480  lb.  wool,  washed;  1600  lb. 
wool,  unwashed;  750  lb.  hides,  green;  2375  lb.  hides,  dry;  128  calf  skins;  46 
muskrat  skins;  256  rabbit  skins;  13  gray  fox  skins;  129  opossum  skins;  382 
raccoon  skins. 

2.  Tabulate  individual  and  gross  weights  of  the  following  order: 

200  16'  I-beams,  weighing  85.2  lb.  per  ft. 
162  19'  I-beams,  weighing  23.5  lb.  per  ft. 
200'  trough  plates,  weighing  16.32  lb.  per  ft. 
1150  18'  channel  irons,  weighing  33  lb.  per  ft. 
450'  deck  beams,  weighing  27.23  lb.  per  ft. 
620  14'  angle  irons,  weighing  17.2  lb.  per  ft. 
145  Z-bars,  12',  weighing  29.8  lb.  per  ft. 

3.  This  is  an  itemized  tabulation  of  the  cost  of  a  masonry  wall  con- 
taining 12,642  cu.  yd.  of  masonry.     Compute  the  total  cost. 

Per  cu.  yd.        Total  cost. 

Quarrying  stone $0,415  $     ? 

Loading  and  hauling  stone 912 

Hoisting  stone 605 

Excavating 300 

Loading  and  hauling  sand 355 

Cement 345 

Mixing  and  dejivering  mortar 205 

Masons  and  helpers 915 

Carpentry 885 

Blacksmithing 190 

Tools  and  general  supplies 175 

Superintendence,  foremen,  etc .725  

(How  check?) 
4 


34  BUSINESS  ARITHMETIC. 

4.     Bids  a,  h,  c,  d  and  e  have  been  submitted  by  different  parties,  ttj 

cover  the  cost  of  a  granite  block  pavement.  Find  and  tabulate  the  tota.« 
cost  of  each  item  and  of  each  bid.  Determine  what  combination  of  dif- 
ferent bids  will  give  the  lowest  cost. 

a  h           c            d          e 

400  cu.  yd.  concrete $4.00  S3.00    $3.90    $2.55     $3.98 

135  lin.  ft.  granite  crossings 1.20  1.13       1.25       1.00       1.24 

1178  lin.  ft.  5"  granite  curb 1.275  1.30       1.30       1.29       1.30 

162  lin.  ft.  circular  curb 2.00  2.10       1.95      2.00      2.0G 

65  lin.  ft.  curb,  reset,  etc 21  .15         .20        .32        .18 

48250  cu.  yd.  grading , 43  .425       .41         .45         .50 

2476  sq.  yd.  granite  paving 3.25  3.18      3.25      3.00      3.30 

2476  sq.  yd.  sub  grade 06  .06         .06         .06        .06 

268  lin.  ft.  24"  dram  pipe 2.78  1.75      2.00      2.10       1.95 

73  cu.  yd.  brick  masonry 12.00  11.25     12.50     10.90     11.75 

21  cu.  yd.  rubble  masonry 5.75  6.00      4.25      5.00      4.80 

12  pc.  Palmer  inlets. 40.00  40.00     40.00    40.00    40.00 

Totals 


CHAPTER  VI. 
DIVISION. 

INTRODUCTORY    EXERCISE. 

1.  Add  by  4s  from  0  to  48.     There  are  how  many  4s  in  48. 

2.  Subtract  by  9s  from  189  to  0.     There  are  how  many  9s  m  189. 

3.  Compare  division  with  addition  and  subtraction. 

4.  I  agree  to  pay  $40  cash  and  $6  per  week  for  a  $100  tjrpewriter. 
Name  the  amount  due  after  each  payment.  Determine  the  number  of 
payments  (1)  by  addition,  (2)  by  subtraction,  (3)  by  division.  Lowering 
the  weekly  payment  to  $4  extends  the  time  of  settlement  ?  weeks. 

5.  172  -^  4  =   ?  ?  X  4  =  172. 

6.  Compare  multiplication  and  division. 

Division  may  be  termed  a  short  method  in  addition  and 
subtraction,  and  the  reverse  of  multiplication.  It  is  the 
process  of  determining  the  unknown  factor  of  a  product  from 
the  product  and  a  known  factor.  The  known  factor  is  the 
divisor,  the  product  is  the  dividend,  and  the  factor  to  be  deter- 
mined is  the  quotient.  The  remainder  is  the  portion  of  the 
dividend  that  is  left  when  division  is  not  exact. 

ORAL    EXERCISE. 

If  the  dividend  is  64  and  the  divisor  8,  show  that: 

1.  The  dividend  divided  by  the  divisor  equals  the  quotient. 

2.  The  quotient  times  the  divisor  equals  the  dividend. 

3.  The  dividend  divided  by  the  quotient  equals  the  divisor. 

FACILITY    EXERCISE. 
(Solve  by  inspection.) 

1.  Divide  by  2:  16,  38,  57,  96,  248,  1749,  62,482. 

2.  Divide  by  3:  12,  21,  48,  69,  125,  683,  952,  667. 

3.  Divide  by  2  and  the  quotient  by  3:  36,  124,  88,  962. 

4.  Divide  by  4:  76,  320,  888,  96,  154,  723. 

35 


36  BUSINESS  ARITHMETIC. 

6.  Divide  by  5:  25,  65,  83,  195,  217,  1259. 

6.  Divide  by  6:  42,  80,  172,  620,  38,  415. 

7.  Divide  by  7:  49,  35,  73^,  1420,  203,  633. 

8.  Divide  by  8:  72,  30,  960,  1240,  884,  1720. 

9.  Divide  by  9:  30,  75,  810,  640,  1850,  27630. 

10.  Multiply  24  and  successive  products  by  2  (6  times).  Divide  by 
successive  4s. 

Short  Division.  In  short  division  the  divisor  and  dividend 
are  written,  and  the  figures  of  the  quotient,  as  obtained,  but 
intermediate  remainders  and  products  are  omitted. 

Illustration.     Divide  7248  by  8. 

Solution.  Analysis.     72  hundreds  -^  8  =  9  hundreds;  4  tens  are 

8)7248  not  divisible  by  8;  4  tens  +  8  units  =  48  units.    48  units 

906  -^  8  =  6  units. 

Note.  At  times,  when  the  divisdr  is  reducible  to  very  simple  factors, 
it  is  possible  to  employ  continued  short  division  by  the  factors  in  turn. 
In  such  cases,  of  course,  each  factor  must  be  an  exact  divisor  until  the  last. 

EXERCISE. 


Divide: 

1.     239,872,452  by  4. 

5.     6888  by  56  (continued  division), 

2.     8,703,265  by  .5. 

6.     24,496  by  16. 

3.     364,285,968  by  6. 

7.     193,625  by  25. 

4.     123,456,789  by  7. 

8.    4,829,672  by  32. 

ILLUSTRATIVE    EXERCISE. 

1.    Solve:  (a)  36  -^  4  =  ? 

(6)  36  ft.  -^  4  =  ? 

(c)  In  36  bbl.  there  are  ?  lots  of  4  bbl.  each. 
2.    What  is  "common"  to  these  problems?    What  changes? 

It  is  essential,  in  division,  to  determine  whether  the  'quotient 
and  the  remainder  are  abstract  or  concrete.  Division  by  an 
abstract  divisor  is  called  partition,  and  by  a  concrete  divisor, 
measuring.  Thus,  in  1  (6)  in  the  preceding  exercise  "  36  ft."  is 
parted  into  4  parts,  and  in  1  (c)  "4  bbl."  is  used  as  a  measure 
to  determine  the  number  of  "4  barrel"  lots  in  36  bbl.     It 


DIVISION.  37 

should  be  noted  that  if  divisor  and  dividend  are  concrete,  the 
quotient  is  abstract;  while  if  the  dividend  is  concrete  and  the 
divisor  abstract,  the  quotient  is  a  concrete  number  of  the  same 
order  as  the  dividend. 

ILLUSTRATIVE    EXERCISE. 

1.  Find  the  remainders  in  the  following: 

(a)  725  ^  4  =  ? 

(6)  420  yd   ^  8  =  ? 

(c)  For  88c  I  can  buy  ?  yd.  of  cotton  at  8c  per  yd. 

2.  When  is  the  remainder  abstract?    When  concrete? 

Long  Division.     In  long  division,  intermediate  products  and 
remainders  are  shown. 

Illustrations.     (1)  Divide  76,945  by  162.  (2) 

Solution.  474  (Quotient)  Table  of  Multiples  of  162. 


Byl 

162 

2 

324 

3 

486 

4 

648 

5 

810 

6 

972 

7 

1134 

8 

1296 

9 

1458 

(Divisor)  162)76945  (Dividend) 
(400  X  162)    648 

1214 
(70  X  162)      1134 
805 
(4  X  162)  648 

157  (Remainder) 
Check.    474  X  162  +  157  =  76945. 


If,  as  often  happens  in  business  and  statistics,  the  same 
divisor  is  constantly  employed,  a  table  of  its  multiples  (see 
(2)  from  1  to  9,  is  prepared,  and  used  for  reference.  This 
avoids  repeated  multiplication,  and  serves  to  show,  without 
trial,  each  successive  figure  in  the  quotient. 

EXERCISE. 

Divide,  expressing  remainder  separately: 

1.  146,732  by  1396.  3.    9,820,605  by  89,243. 

2.  $372,486  by  2013.  4.    620,548  lb.  by  63  lb. 
5.     In  468,246  lb.  of  grain  there  are  ?  bu.  of  56  lb.  each. 


38  BUSINESS  ARITHMETIC. 

6.  Divide  the  following  by  1237,  using  a  table  of  multiples: 

(a)  673,892.  (6)  960,478.  .  (c)  1,214,734. 

7.  Divide  the  following  by  864,  using  a  table  of  multiples: 

(a)  $4,327,432.        (b)  8,921,065.  (c)  1,234,056,789. 

8.  Find  the  number  of  tons  of  coal  (2240  lb. )  in  each  of  these  shipments, 
ignoring  fractions  of  a  ton. 

Car  No.  Wt.  of  Loaded  Car.  Empty  Car.  Net  Weight.  Tons. 
32,064                    134,200                         26,420                         ?  ? 

38,296  129,100  27,020 

43,129  126,000  24,320  ?  ? 

Short  Methods. 

Division  does  not  lend  itself  to  so  many  easily  applied  short 
methods  as  does  multiplication.  The  following  are  the  most 
common : 

Division  by  10 y  100 ^  etc. 

ORAL    EXERCISE. 

1.  What  effect  has  the  decimal  point  on  the  comparative  values  of 
these  numbers:  6800,  680  0,  68.00,  6.800? 

2.  Suggest  a  short  method  of  division  by  powers  of  10.  Find  the 
quotient  of: 

3.  720,000  ^  10.  6.    30,000,000  lb.  ^  10,000. 

4.  726,000  -^  1000.  7.     368,000  ^  100. 

5.  $725,900   -h  $100.  8.     $850  -^  10. 

Division  by  multiples  of  10.  When  a  power  of  ten  is  a  factor 
of  both  dividend  and  divisor,  the  example  may  be  simplified  by 
the  principle  of  continued  division. 

Illustration.    4720  -^  80  =  (4720  -^  10)  -^  8  =  472  4-  8  =  59. 

EXERCISE. 

Find  the  quotient  by  inspection. 

1.  840  ^  70.  4.    320  lb.  -=-  20.  7.    $650  ^  130. 

2.  725,000  -^  600.   5.  $7,600  ^  40.       8.  $987,500  -J-  500. 

3.  870  -^  30.      6.  800,000  lb.  -^  40. 


DIVISION.  39 

DECIMAL  DIVISION. 
INTRODUCTORY    EXERCISE. 

1.  4.2  is  equivalent  to  ?  tenths 

2.  4.2  -i-  6  may  be  expressed:  ?  tenths  -^  6.    The  quotient  is  — . 

3.  Divide  by  6:  $480,  48  lb  ,  48  T.,  .048  cu.  yd.,  .00048. 

4.  Divide  these  same  values  by  .6.  What  remains  unchanged  in  the 
quotient?    What  changes? 

It  is  evident  that  the  location  of  the  decimal  point  in  the 
quotient  is  the  new  feature  of  the  division  process  met  with 
in  passing  from  whole  numbers  to  decimals. 

Illustrations.     (1)  Divide  49.776  by  16. 


Hon 

Condensed  solution. 

3111 

3,111 

16)49.776 

16)49.776 

(3  X  16) 

48 

,      -          48 

1  776  remaining. 

17 

(.1  X  16) 

16 

16 

.176  remaining. 

17 

(.01  X  16) 

.16 

16 

.016  remaining. 

16 

(.001  X  16) 

.016 

16 

Note.  If  the  divisor  is  integral,  the  position  value  of  each  figure  in 
the  quotient  is  the  lowest  position  value  in  the  partial  dividend  used  in 
obtaining  it.  Thus  "49"  is  the  partial  dividend  used  in  obtaining  "3" 
of  the  quotient,  and  its  lowest  position  value  is  ''unity."  The  advantage 
of  writing  quotient  over  dividend  is  evident,  since  the  "placing"  of  the 
first  figure  of  the  quotient  practically  determines  the  place  value  of  the  rest. 

(2)  Divide  7.6845  by  .15. 

Solution.     Multiply  each  term  by  100,  by  short  method. 

7.6845  H-  .15  =  768.45  -^  15. 
•  51.23 

15)768.45 

Note.  In  case  the  divisor  is  decimal  in  form,  it  may  be  cleared  of 
decimals,  by  mo^dng  the  decimal  point  to  the  right  in  both  dividend  and 
divisor.  The  example  then  comes  under  the  class  shown  in  the  previous 
illustration. 


40  BUSINESS   ARITHMETIC. 

EXERCISE. 

Using  short  methods,  if  convenient,  find  by  inspection  the  quotient  of; 

1.  3.6^6.  4.    .0004^2.       7.    3.824-r-8.  10.    52.08  T.-^100. 

2.  .427-^7.         5.    2.644-3.         8.    16.860-J-12.      11.    3.62-^10. 

3.  .084-^16.       6.    1.64-4.  9.    .00129-^30.      12.    29000.01 -MOOO. 
In  the  following,  clear  the  divisor  of  decimals: 

13.    .844-1.2.         14.    63.08-i-.002.     15.  .00034-  006.      16.    2.40604-1.2. 
Divide,  by  inspection: 

17.  1  by  .01,  .001,  .0001,  10000.  22.  200,  4000,  .08  and  .0006  by  2. 

18.  .01  by  10,  100,  .01  and  .00001.  23.  .00012  by  6,  600,  .3,  .03,  .006. 

19.  .0001  by  .005,  .05,  .02,  100.  24.  4.24  by  .0006  and  by  6. 

20.  4.8  by  1.2,  .012,  .0006,  .3.  25.  63  by  .007,  .09,  .7,  .0003. 

21.  60  by  .002,  .3,  .0005,  .12.  26.  660.15  by  .015. 

Exact  and  Approximate  Results.  Decimal  ciphers  may  be 
added  to  the'  dividend  indefinitely,  to  secure  an  accurate 
quotient  to  any  number  of  decimal  places,  the  final  remainder 
being  expressed  fractionally.  Commonly,  however,  results 
are  required  exact  to  some  stated  number  of  places,  the  remainder 
being  unstated.  Thus  a  business  man  may  compute  costs 
"to  the  nearest  cent,"  or  an  engineer  make  measurements  to 
"hundredths  of  an  inch."  Many  working  tables,  also,  are 
computed  to  four,  five  or  six  decimal  places,  remainders  being 
ignored  because,  as  a  rule,  they  are  too  small  to  affect  appre- 
ciably results  computed  by  them.  Often,  if  the  remainder  is 
one-half  or  more  of  the  preceding  decimal  figure,  that  figure 
is  increased  by  1.  Thus  a  quotient  of  7.3086  may  be  written 
"to  three  decimal  places"  as  7.309. 

When  merely  an  approximate  quotient  is  desired,  the 
approximate  method  is  sometimes  used.  This  consists  usually 
in  direct  division  until  the  integral  portion  of  the  quotient  is 
obtained.  From  that  point,  no  further  figure  of  the  dividend 
is  brought  down  but,  at  each  step,  the  right  hand  figure  of 
the  previous  divisor  is  cancelled. 


DIVISION. 


41 


Illustration.    Divide  684.394683  by  56.138. 
Contracted  Soluiion. 
12.1914 
56.138)684.3946839 
56138 
123  014 
112  276 
10  738 
(Div.  56.13)         5  613 
5  125 
(Div.  56.1)  5  049 

76 
(Div.  56)  56 

20 
(Div.  5)  20 

EXERCISE. 

Divide: 

1.  $726  48  by  12.  4.     .0297  by  .31.     (2  places  ) 

2.  114,269  lb.  by  11.     (2  places.)     5.     12.873  by  151.3.     (1  place.) 

3.  4629.2  cu.  yd.  by  .24.  6.     842,097  by  .0036.     (Approx.) 

7.  23.0567385  by  .07525.     (3  places  approximate.) 

8.  $7296.47  by  123.     (3  places.) 

9.  91.38  by  1.02545.     (3  places,  approximate.) 

10.  A  speed  of  592  knots  in  24  hours  is  a  speed  of  ?  knots  per  hour. 
(2  places.) 

EXERCISE. 

Note.  The  weight  of  any  material,  as  compared  with  an  equal  bulk 
of  pure  water,  is  termed  its  specific  gravity.  Thus  a  brick  having  a  specific 
gravity  of  1.602  weighs  1.602  times  as  much  as  the  same  volume  of  water. 

Assuming  the  weight  of  a  cubic  foot  of  water  as  62.355  lb.,  find  the 
specific  gravity  of  the  following  building  materials: 

Material.  Pounds  per  Cu.  Ft.  Specific  Gravity. 

Asphaltum 87 

Hard  brick 125  

Clay 120  to  145  to 

Limestone 170  to  210  to 

Plaster  of  paris 75  to    80  to 

Slate 165  to  180 to 

Tile 106  to  122  to 

Sand  stone 140  to  155  to ■ 


42  BUSINESS  ARITHMETIC. 

Selected  Individual  Assignments. 

1.  Write  a  brief  on  "Methods  of  Checking  Arithmetical  Computa- 
tions/' giving  illustrative  examples. 

2.  Write  a  brief  on  "Short  Methods  in  Computation."  Illustrate 
short  methods  in  common  use.  If  possible,  report  on  business  men's 
attitude  towards  the  use  of  short  methods. 


CHAPTER  VII. 
ARITHMETICAL  AVERAGING. 

INTRODUCTORY    EXERCISE. 

1.  What  six  equal  numbers  have  a  total  equal  to  the  sum  of  12,  14, 
30,  42,  18  and  10? 

2.  Five  men  give  to  a  public  cause,  respectively,  $40,  $35,  $20,  $60,  $25. 
The  same  sum  would  have  been  secured  by  their  contributing  equally,  $ — . 

3.  The  total  profit  of  a  business  for  three  years  was  $19,654.21.  As- 
siuning  that  there  was  no  variation  from  year  to  year,  what  was  the  annual 
profit  ? 

4.  The  values  just  determined  are  called  averages,  or  average  numbers. 
What  is  an  average  number?  What  processes  of  arithmetic  are  employed 
in  averaging? 

5.  What  does  a  physician  mean  when  he  says  "My  office  calls  average 
12  per  day"? 

Averages  offer  a  common  and  effective  means  of  comparing 
values,  and  are  used  constantly  in  statistical  work  and  in 
business.  Thus  the  average  output,  expense  and  profit  of  a 
business,  for  past  years,  will  guide  a  possible  purchaser  to  a 
decision,  or  aid  an  owner  in  remedying  defects.  The  average 
monthly  rainfalls,  in  a  certain  farming  section,  will  guide  an 
intending  purchaser  in  his  decision  as  to  possible  crops.  Other 
illustrations  will  be  found  throughout  the  book. 

Ordinarily,  the  average,  mean,  or  normal,  is  obtained  by  a 
process  of  addition,  followed  by  division.  At  times,  the  addi- 
tion may  be  mechanically  accomplished,  as  in  the  case  where 
a  number  of  castings  are  placed  on  a  foundry  scale  and 
weighed  at  one  operation,  the  total  weight  being  then  divided 
by  the  number  of  castings  to  determine  the  average  weight. 

In  determining  averages  care  must  be  used  to  include  all 
influencing  factors. 

.    43 


44  BUSINESS   ARITHMETIC. 

Illustration.  On  April  1,  Brown  deposits,  in  a  bank  that  pays 
interest  on  average  monthly  balances,  the  sum  of  $2000;  a^d  on  April  21, 
reduces  this  sum  lo  $1600.     Determine  his  average  balance  for  April. 

Solution.  The  average  is  not  $2000  +  1600  divided  by  2,  for  the 
individual  balances  existed  for  different  times.  $2000  was  the  balance 
for  20  days,  and  $1600  of  10  days.  There  were  20  daily  balances  of  $2000 
and  10  of  $1600.     The  average  is,  therefore, 

20  X  $2000  4-  10  X  $1600     $56000     ^.^^^^^ 
30 ="^0~  =^1866.67  -. 


EXERCISE. 

Note.  These  sentences  are  taken  at  random  from  news  reports  and 
general  publications.     Determine  the  missing  values. 

1.  "The  Lusitania  made  598,  606,  612  and  603  knots  in  four  successive 
days,  averaging  —  knots  per  day." 

2.  "England  records  a  West  Coast  train  from  London  to  Aberdeen, 
September,  1895,  540  miles  in  512  minutes miles  per  hour." 

3.  "In  1895  the  number  of  depositors  in  the  savings  banks  of  the 
country  was  7,696,000,  having  total  deposits  of  $3,093,236,119,  or  an 
average  of each." 

4.  "The  distance  is  192  miles,  but  I  can  average  12  miles  an  hour.  If 
I  leave  at  7  o'clock  to-morrow,  that  would  get  me  there  by -" 

5.  "Those  cattle  ought  to  average  940  lb.  each,  or lb.  for  the 

lot  of  37." 

6.  "The  output  with  the  old  machinery  averaged  46,000  yd.  per  10 
hour  day.  With  the  new  equipment,  it  is  48,000  yd.  in  9  hours.  The 
average  hourly  output  has  increased yd." 

7.  "Mr.  Newbold  testified  that  the  improvement  had  increased  their 

average  daily  output  from  143.2  tons  to  158.5  tons — an  increase  of 

tons  per  year."     (1  year  =  300  days.) 

EXERCISE. 

Note.    Each  example  shows  a  different  application  of  the  average. 

1.  In  a  history  test,  the  individual  marks  obtained  were:  47,  90,  87, 
82,  100,  100,  96,  90,  40,  35,  82,  87,  or  a  class  average  of—.     (2  dec.  places.) 

2.  Twenty  pupils  in  a  class  were  present  on  the  twenty-one  school 
days  in  March,  four  were  absent  one-half  day,  three  for  one  day,  two  for 
three  days,  and  one  for  five  and  one-half  days.  Find  the  average  daily 
attendance. 


AVERAGING.  45 

3.  In  an  Eastern  city,  having  a  normal  maximum  temperatm-e  for 
July  of  76.5"  a  hot  wave  caused  maxima,  for  successive  days,  of  90°,  93°, 
95.5°,  89°,  96°,  98°,  101°,  92°,  an  average  of  — °  above  normal. 

4.  During  November,  1907,  individual  rainfalls  were  .45  in.,  2.15  in., 
.08  in ,  1.37  in.,  and  .78  in.  Was  the  rainfall  excessive  as  compared  with 
(4.88  in.)  the  normal  for  past  years? 

5.  The  Rankin  Mfg.  Co.  pays  60  men  $2.15  per  day;  80  men,  $3  00; 
12  men,  $4.50;  and  8  men,  $5.75.  How  does  its  average  daily  wage  compare 
with  that  of  the  Morgan  Co.,  which  has  an  average  for  the  same  class  of 
work  of  $3,225? 

6.  Find  the  average  daily  and  monthly  circulation  of  the  newspaper 
whose  circulation  statement  is  printed  on.  page  12. 

7.  Find  the  average  speed  on  these  long-distance  runs: 

Course.  Distance.        Hours  and  Min.        Average. 

Jersey  City  and  Chicago 912  mi.  17:41 

Jersey  City  and  Pittsburgh ...  443  8:41 

New  York  and  Buffalo 440  8:15 

New  York  and  Chicago 979.52  18:00 

London  and  Edinburgh 393  7:45 

London  and  Glasgow 401.50  8:00 

Paris  and  Bayonne 486.25  8:59 

8.  Compare  the  profits  of  the  company  mentioned  below  for  the  last 
year  with  the  average  for  the  past  five  years. 

Net  Profits  of  the  Cropleigh  Wheel  Company. 

1906  $17,246  1908        $18,076.50  1910        $20,132.61 

1907  21,032  1909  16,548  1911  23,328.46 

1912  24,736.59 

9.  Explain  the  difference  in  meaning  of  these  expressions: 

(a)  "126  pieces  of  castings,  averaging  84  pounds  each." 
(6)  "  126  pieces  of  casting  weighing  92  pounds  each." 
(c)  "45  pieces  of  castings  weighing  2746  pounds." 


CHAPTER  VIII. 
THE  EQUATION. 

INTRODUCTORY    EXERCISE. 

1.  Use  the  word  "equal"  in  comparing  the  cost  of  two  articles. 

2.  Use  the  same  word  in  expressing  the  cost  of  one  article. 

3.  Use  it  in  comparing  the  time  necessary  to  do  two  pieces  of  work. 

4.  Use  the  word  in  expressing  the  time  it  takes  to  go  from  Philadelphia 
to  New  York,  by  rail. 

5.  Use  it  in  comparing  the  size  of  two  objects,  one  greater  by  one-half 
than  the  other. 

An  equation  is  formed  by  using  the  equality  sign  (=)  be- 
tween two  equal  numbers  or  expressions,  called  the  members  or 
sides  of  the  equation. 

Illustrations. 

My  business  for  May  =  that  for  April. 

The  cost  of  5  bbl.  at  $6  =  the  cost  of  6  bbl.  at  $5. 

The  quotient  X  the  divisor  =  the  dividend. 

Certain  self-evident  truths  (axioms)  aid  us  in  the  practical 
use  of  the  equation: 

1.  Things  equal  to  the  same  thing  are  equal  to  each  other. 

2.  The  equality  of  two  expressions  is  not  destroyed: 
(a)  By  adding  equals  to  each  term. 

(6)  By  subtracting  equals  from  each  term. 

(c)  By  multiplying  each  term  by  the  same  factor. 

{d)  By  dividing  each  term  by  the  same  factor. 

Illustrations. 

1.  If  one  barrel  of  apples  cost  $3  and  one  barrel  of  sweet  potatoes 
costs  $3,  then — 

The  cost  of  1  bbl.  apples  =  the  cost  of  1  bbl.  sweet  potatoes. 
If  a  =  6  and  6  =  6,  then  a  =  6. 

46 


THE   EQUATION.  '  .  47 

2.  If  the  cost  of  1  bbl.  apples  =  cost  of  1  bbl.  sweet  potatoes,  then 

{a)  Cost  of  1  bbl.  apples  +  $4  =  cost  of  1  bbl.  potatoes  +  S4. 
(6)  Cost  of  1  bbl.  apples  —  $1  =  cost  of  1  bbl.  potatoes  —  $1. 
(c)  Cost  of  6  bbl.  apples  =  cost  of  6  bbl.  potatoes. 
,r^  Cost  of  6  bbl.  apples      cost  of  6  bbl.  potatoes 
id)  2 = 2 • 

3.  If  a  =  6,  then — 

(a)  a  +  X  =  b  -\-  X. 
(]b)  a-y  =h'-y, 

(c)  6a  =  66. 

(d)  a/4  =  6/4. 

Transpositicm.  A  term  in  an  equation  may  be  changed  from 
one  side  to  the  other  provided  its  algebraic  sign  is  changed 
from  plus  to  minus,  or  from  minus  to  plus. 

Illustration. 
Ifx+I0  =  y 

a;  +  10  -  10  =  ?/  — 10.     (Subtract  10  from  each  side.) 
X  =  y  —  10.    That  is,  the  +  10  on  the  left  has  been  changed  to  a 
—  10  on  the  right. 

In  a  similar  way,  the  equation  "  a—b=c"  may  be  written  "a=c+6." 

If  in  an  equation  there  occurs  an  expression  whose  value 
is  not  given,  the  value  may  be  determined,  in  most  cases,  by 
transposition.     This  is  termed  solving  the  equation. 

Illustration. 

Let  a  +  6  =  9  +  4,  to  find  the  value  of  a;  then 

a  =  9  +  4  —  6  (transposing  6),  and 

a  =  7. 

In  handling  "signs,"  it  is  essential  to  have  their  order  of 
rank,  or  precedence,  understood.  Of  the  four  signs,  +,  — ,  X 
and  -^ ,  the  multiplication  and  division  signs  have  precedence 
over  those  of  addition  and  subtraction. 

Illustration. 

Let  a  =«  52  —  6  X  8.  The  multipUcation  sign  is  considered  first,  and 
we  have  a  =  52  —  48  =  4. 

The  signs  X  and  -i-  have  equal  weight,  and  terms  contain- 
ing them  are  reduced  to  simpler  form  by  performing  each 


48  .  '  BUSINESS  ARITHMETIC. 


operation  as  its  sign  is  met.     The  signs  of  addition  and  sub- 
traction also  have  equal  weight. 

Illustrations. 

9  X  8  4-  3  =  72  -i-  3  =  24. 

4  +  7-3  =  11 -3  =  8. 

Note.  The  signs  of  addition  and  subtraction  refer  to  all  following 
values  until  the  next  plus  or  minus  sign.  Thus  in"3+8X6-^2+  8," 
the  *'  +  "  refers  to  the  result  of  "8  X  6  -r  2."  The  expression  equals 
3  +  48/2  +  8  =  35. 

•  EXERCISE. 

1.  Illustrate  the  five  axioms,  using  the  equation  a  +  8  =  6  —  4. 

2.  Find  the  "cost"  from  the  equation  "The  cost  of  a  house  +  $275 
=  $8450  -  $90." 

3.  Find  the  value  of  1  A.  from  the  equation:  Cost  of  21  A.  =  $640.50 

4.  Solve  for  ar,  if  a;  -  70  +  4  =  20  X  6  -^  5. 

5.  Dividend  X  8  =  64.     What  is  the  dividend? 

6.  Solve:  2+4-7  =  aX6-^2-5. 

7.  A  number  +  16  =  128.     Find  the  number. 


CHAPTER  IX. 

UNITED  STATES  MONEY. 

INTRODUCTORY    EXERCISE. 

1.  What  is  your  own  definition  of  a  coin? 

2.  What  denominations  of  coin  have  you  handled?  What  others 
can  you  name? 

3.  Read  the  wording  on  a  one-dollar  bill.  What  is  the  difference 
between  it  and  a  silver  dollar?    What  makes  the  paper  money  of  value? 

4.  WTiat  causes  the  difference  in  size  between  a  silver  dollar  and  a  gold 
quarter  eagle? 

5.  WTiy  should  not  the  Government  coin  iron,  as  money,  in  place 
of  gold? 

6.  In  what  ways  would  a  man  be  inconvenienced,  who  had  much 
property  but  absolutely  no  money?  Could  he  get  along  comfortably 
without  money? 

7.  What  is  the  disadvantage  in  trading  one  kind  of  property  for 
another — instead  of  using  actual  money  to  pay  for  the  desired  article? 

One  function  of  a  modern  government  is  to  provide  a  system 
of  money  or  currency  to  serve  as  a  "  medium  of  exchange.'*  A 
primitive  people  may  use  anything  as  a  standard,  as  corn, 
wampum,  horses.  As  commerce  grows,  and  one  who  desires 
an  article  cannot  meet  directly  a  distant  producer  who  has  it, 
property  must  pass  through  several  hands;  the  primitive 
property  standards  then  become  difficult  of  use,  and  also 
subject  the  users  to  loss  and  fraud.  Then  the  government,  as 
a  representative  of  all  the  people,  adopts  as  standard  something 
easily  transferable,  and  puts  its  mark  upon  it  with  such  care 
and  in  such  a  way  that  to  every  holder  a  certain  quantity  may 
represent  the  same  value.  Thus  gold  is  the  standard  of  this 
country,  and  the  coming  standard  of  the  world,  because  high 
values  are  imputed  to  small  quantities  of  it,  making  it  easy 
of  transfer,  and  because,  when  coined,  exact  uniformity  of 
5  49 


50  BUSINESS  ARITHMETIC. 

weight  and  value  may  be  given  artificially  to  each  piece. 
Small  denominations  are  coined  in  metals  of  less  value  than 
gold,  because  otherwise  the  coin  would  be  too  small  to  handle 
and  it  would  be  easily  lost.  Such  coin,  in  suflficient  sums,  may 
be  exchanged  for  gold.  Likewise,  principally  for  convenience 
in  transferring  large  sums,  paper  certificates  are  issued  in  ex- 
change for  gold  and  silver  and  are  payable  on  demand  in  coin. 
The  currency  now  in  use  in  the  United  States  consists  of  metal 
coins,  and  paper  notes  and  certificates. 

UNITED  STATES  CURRENCY. 
Metal. 

Gold.                                Silver.  Nickel. 

Double  eagle .  .  $20.00  Dollar $1.00        Nickel $.05 

Eagle 10.00  Half  dollar 50  Brmize. 

Half  eagle 5.00  Quarter  dollar. .     .25  Penny   (one  cent 

Quarter  eagle. .     2.50  Dime 10  piece) 01 

Paper. 

1.  Gold  certificates.  Denominations:  $20,  upward  to  $20,- 
000 .  Issued  for  gold  deposited  in  the  United  States  Treasury, 
and  payable  in  gold  on  demand. 

2.  Silver  certificates.  Denominations:  $1.00,  $2.00,  $5.00, 
$10.00,  $20.00,  $50.00,  $100.00.  Issued  for  silver  deposited 
in  the  Treasury,  and  payable,  on  demand,  in  the  same  metal. 

3.  United  States  notes  (Greenbacks).  Denominations: 
$10,  $20,  $50,  $100,  $500,  $1000.  They  are  a  part  of  the  debt 
of  the  United  States  and  consist  of  promises  to  pay  in  coin  the 
number  of  dollars  specified  on  demand. 

4.  National  bank  notes.  Denominations  $5,  and  upward  to 
$1000.  They  are  issued  by  national  banks,  under  government 
supervision,  and  are  redeemable  in  lawful  money. 

Note.  Legal  tender  is  a  term  applied  to  money  that  may  be  legally 
ofiFered  in  payment  of  debts.  Gold  and  silver  coin  of  standard  weight,  and 
gold  and  silver  certificates,  are  legal  tender  for  all  debts.     United  States 


UNITED   STATES   MONEY. 


51 


and  national  bank  notes  are  legal  tender,  except  for  duties  on  imports  and 
interest  on  the  national  debt. 

The  Government  has  five  mints,  or  coin  factories,  where  gold 
or  silver  bullion  is  received  for  coinage,  and  seven  assay  offices. 
The  assay  offices,  as  well  as  the  mints,  receive  deposits  of  gold 
and  silver,  and  also  manufacture  refined  bars  of  standard 
purity.  The  bars,  when  returned  to  the  depositors,  are  used  in 
the  arts,  or  are  exported.  The  metal  left  in  bars  far  exceeds 
in  value  the  metal  coined.  The  mints  are  situated  at  Phila- 
delphia, San  Francisco,  New  Orleans,  Carson  City  and 
Denver.  The  assay  offices  are  scattered  throughout  the 
country  at  the  main  points  of  metal  production  and  importa- 
tion. Some  idea  of  the  volume  of  business  may  be  obtained 
from  these  figures  for  a  recent  year: 


Coinage  of  the  United  States. 


Denomination. 

Pieces. 

Value. 

Gold 

Double  eacles         .  • 

1  495  035 

1  494  795 

1  559  435 

142  509 

$     ? 

Eagles 

Half  eagles 

Quarter  eagles 

Total              

? 

1  621  700 

4  422  700 

20  998  431 

? 

Silver. 

Half  dollars 

? 

Quarter  dollars                          

Dimes 

Total                      

? 

46  047  950 

89  588  480 

? 

Minor. 

Nickels 

? 

Pennies 

Total 

? 

? 

Total  Coinage 

? 

? 

EXERCISE. 

Find  the  missing  values  in  this  and  the  following  tables. 

Worn  or  abraded  coins  are  collected  through  the  banks  and 


52 


BUSINESS   ARITHMETIC. 


are  recoined.     The  loss  through  abrasion  is  considerable,  as 
the  following  table  shows: 


Recoinage  of  Uncurrent  Coin. 
(Six  recent  years.) 

Face  Value. 

New  Coin. 

Loss. 

1 

2 
3 
4 
5 
6 

$3  832  280.69 
3  333  437.06 
3  008  747.98 
2  828  384.90 
1964  476.11 
1  414  963.90 

$3  613  021.59 
3  141  548.04 
2  829  890.71 
2  656  104.21 
1  839  219.24 
1  322  834.27 

? 

Total,  6  yr. 
Average 

?                                       ? 
?                                       ? 

While  the  total  amount  of  money  in  circulation  seems  very 
large,  the  actual  amount  of  money  in  circulation  per  individual 
of  our  population  seems  remarkably  small,  and  shows  to  what 
extent  we  carry  on  transactions  by.  means  of  checks  and  other 
forms  of  credit,  without  actually  handling  currency. 


Money  in  Circulation. 
(Five  recent  years.) 


Population. 


84,662,000 
86,074,000 
87,496,000 
88,926,000 
90,363,000 


Money  in  Circulation. 


$2,736,646,628 
2,772,956,455 
3,038,015,488 
3,106,240,657 
3,102,355,605 


Circulation 
per  Capita. 

$32  32 
? 
? 
? 
? 


IS  a 


As  has  already  become  evident,  the  "money  value" 
common  factor  in  a  large  part  of  the  arithmetic  calculations 
incident  to  our  daily  life,  yet  in  the  majority  of  cases  the 
actual  calculations  are  based  on  dollars  and  decimals  thereof — 
and  no  knowledge  of  denominations  is  essential  to  solutions. 
The  following  classification  of  United  States  money  is  common, 


UNITED   STATES    MONEY.  53 

although  the  dime  and  eagle  are  usually  omitted,  being  con- 
sidered simply  the  names  of  coins. 

Table  of  United  States  Money. 
10  mills      =  1  cent.  Expressed  c,  ct.,  ^,  $.01. 

10  cents     =  1  dime.*  "         d.  or  $.10. 

10  dimes    =  1  dollar.  "  $1.00. 

10  dollars  =  1  eagle.  "         E. 

When,  as  a  result  of  a  calculation  (as,  for  example,  the  deter- 
mination of  cost  or  wage)  a  payment  of  money  must  be  made, 
knowledge  of  denominations  of  coins  and  certificates  becomes 
of  distinct  value.  A  cashier,  a  bank  teller,  or  a  pay  clerk, 
must  have  extended  knowledge  even  of  issues  of  coins  and 
notes  and  of  the  minute  details  that  distinguish  the  genuine 
from  counterfeits.  Moreover,  such  employees  must  be  able 
to  make  change  quickly  and  must  be  able  to  plan  out  and 
arrange  for  the  quantities  of  coins  of  different  denominations 
necessary  during  each  day  for  the  easy  transaction  of  business. 

In  connection  with  the  cash  payments,  many  notes  and 
checks  are  handled.  These,  and  similar  papers,  usually  have 
the  money  value  stated  twice,  once  in  words,  as  briefly  as  pos- 
sible, and  once  in  figures.  Portions  of  a  dollar  are  usually 
expressed  in  figures,  fractionally.     See  page  323. 


CHAPTER  X. 
MAKING  CHANGE. 

Speed  and  accuracy  in  change  making  and  in  making  money 
payments  are  essential  in  an  increasing  number  of  positions. 

Making  change  is  a  process  of  addition.  The  cashier  adds 
mentally  to  the  amount  of  the  purchase  each  coin  or  note  he 
takes  from  his  tray,  until  he  reaches  the  total  of  the  sum  given 
in  payment.  In  most  cases  he  does  not  know  the  exact 
amount  of  the  change  returned. 

iLTiUSTRATioN.  A  $10.00  note  is  given  in  payment  for  a  purchase 
of  $2.37.  The  cashier  takes  from  his  tray  three  pennies,  a  dime,  a  half- 
dollar,  a  two-dollar  note  and  a  five-dollar  note,  saying  aloud,  if  dealing 
with  a  customer,  "Thirty-eight,  thirty-nine,  forty,  fifty,  three  dollars, 
five,  ten  dollars." 

Question.    Why  is  this  better  than  a  subtraction  method? 

The  careful  salesman,  on  receiving  a  note,  states  its  value 

to  the  one  who  gave  it  to  him,  before  sending  it  to  the  cashier, 

in  order  to  avoid  argument  when  the  change  is  returned.     In 

the  above  case,  the  salesman  might  remark,  "You  gave  me 

$10.00." 

Question.  State  some  simple  misunderstanding  that  might  arise  from 
neglect  of  this  precaution. 

ORAL   EXERCISE. 

Note.  In  practising  change-making,  name  the  sub-totals  until  the 
amount  is  reached.  When  not  reciting,  watch  the  totals  given  to  determine 
the  currency  denominations  used.     Suggest  other  combmations. 

1.  Name  the  common  United  States  coins,  certificates  and  notes. 

2.  Make  change  for  the  foUowmg: 


irchase, 

.  Pavment. 

Purchase.   Payment. 

Purchase.  Payment. 

$5.47., 

. .  .$io.oo 

$15.43....  $20.00, 

$.01.... $.10 

3  19. 

...  20.00 

.08 25 

.18 50 

.67., 

.    .     2.00 

1.32....     5.00 

3.21....  3.50 

.11., 

. .  .       .50 

.21 50 

.83...,  5.00 

1.14. 

...     5.00 

.43...     2.00 
64 

2.17.... 4.00 

MAKING   CHANGE.  55 

3.  In  each  of  the  above  cases,  name  the  denomination  of  coin  or  notes 
given  in  payment. 

4.  Why  should  not  a  customer  give  a  salesman  $4.27  for  a  purchase  of 
$3.09? 

Make  change  for: 

5.  $10.00,  given  in  payment  for  two  chairs  @  $1.17. 

6.  $5.00,  in  payment  for  May  gas  bill,  $2.53. 

7.  $10.00,  in  payment  for  one  pair  of  shoes,  $3.75. 

8.  50c,  in  payment  for  17  Ic  stamps. 

Note.     Cashiers  may  have  to  break  notes  into  smaller  denominations, 
or  to  make  specified  payments. 

9.  Change  a  $1  00  bill  into  nickels  and  dimes. 

10.  "  Break  up  "  a  $10.00  bill. 

11.  Pay  a  check  for  $124,  in  denominations  of  $5.00  and  under. 

12.  Count  out  $15.42  for  a  pay  envelope. 

13.  Pay  your  milk  man,  $3.41. 

14.  Pay  John  Brewer  $700  in  Treasury  notes. 

15.  Pay  John  Smith  $445  in  gold  coin. 

FOR    DISCUSSION. 

1.  Do  all  businesses  require,  for  change-making,  the  same  proportions 
of  coins  of  each  denomination? 

2.  Inquire  of  employees  of  ten  distinctive  businesses  what  coins  are 
most  needed  for  change-making.     Report  to  class. 

3.  What  businesses  receive  more  change  than  they  need? 

4.  Name  businesses  receiving  large  quantities  of  pennies,  nickels  and 
dimes. 

5.  Find  out  the  proportions  of  coins  required  by  a  commercial  bank 
in  meeting  the  demands  of  its  customers  for  change. 

6.  At  what  seasons  of  the  year  do  banks  make  special  preparations  for 
supplying  small  change  and  bills? 

7.  Investigate  and  report  on  money  machines,  such  as  coin  trays, 
cash  registers,  change-making  machines,  etc. 


CHAPTER   XI. 
POSTAGE. 

INTRODUCTORY    EXERCISE. 

1.  Is  postage  a  tax? 

2.  Are  stamps  used  on  all  mail  matter? 

3.  Why  do  the  stamp  denominations  differ  in  part  from  our  coin 
denominations? 

4.  What  government  stamps  are  issued  for  other  objects  than  the 
direct  payment  of  postage? 

5.  Name  some  limitations  as  to  acceptable  mail  matter? 

6.  How  does  the  modern  postal  service  help  business? 

7.  What  is  "rural  free  delivery"? 

8.  Why  is  the  regular  postage  rate  in  a  country  the  same,  irrespective  of 
distance  the  matter  is  carried? 

9.  TMiat  is  the  "  parcels  post "? 

The  Post  Office  Department  is  one  of  the  greatest  of  business 
organizations  under  government  control.  It  is  not  run  for 
profit,  but  its  enormous  expenses  are  met,  largely,  by  postage 
and  money  order  receipts.  Some  idea  of  its  cost  may  be 
obtained  from  these  figures  for  five  recent  years: 

Post  Offices  and  Post  Routes. 


Year. 

Post 
Offices 
Num- 
ber. 

Extent 
of  Post 
Routes 
Miles. 

Revenue  of 

Department 

Dollars. 

Expended  for  Trans- 
portation. 

Total  Ex- 
penditure of 
Department. 

Defi- 

Domestic 
Mail. 

Foreign 
Mail. 

cit. 

1 
2 
3 
4 
5 

65,600 
62,659 
61,158 
60,144 
59,580 

478,711 
463,406 
450,738 
448,618 
447,998 

167,932,782 
183,585,006 
191,478,663 
203,562,383 
224,128,637 

72,944,352 

77,471,917 
78,174,988 
80,901,899 
81,709,433 

2,895,756 
2,988,849 
2,982,732 
2,804,170 
3,164,254 

178,449,779 
190,238,288 
208,351,886 
221,004,102 
229,977,224 

Total 

? 

? 

? 

? 

56 


POSTAGE.  57 

POSTAL  INFORMATION. 
1.     Domestic  Mail  Matter. 

Postage  rates  apply  also  to  mail  matter  for  Canada,  Canal  Zone,  Cuba, 
Guam,  Hawaii,  Mexico,  Philippines,  Porto  Rico,  Republic  of  Panama, 
Tutuila,  and  the  U.  S.  postal  agency  at  Shanghai.  The  domestic  rate, 
for  letters  only,  applies  to  Germany,  Great  Britain  and  Ireland,  and 
Newfoundland . 

First  Class.  Letters  and  sealed  matter:  2c  for  each  ounce  or  fraction. 
Postal  cards  and  post  cards:  Ic  eacL     Limit  of  weight,  4  pounds. 

Second  Class.  Newspapers  and  periodicals:  Ic  per  4  ounces  or  fraction. 
No  weight  Umit.     Publisher's  rate,  Ic  per  pound. 

Third  Class.  Books  and  miscellaneous  printed  matter:  Ic  per  2  ounces 
or  fraction.     Limit  of  weight,  4  pounds,  except  for  a  single  book. 

Fourth  Class.  (Parcels-Post.)  All  matter  not  in  other  classes.  Seeds  and 
plants,  Ic  for  2  oz.  For  other  matter  the  coimtry  is  divided  into  unit 
areas  and  ei^t  zones.  The  first  zone  includes  territory  within  a  radhis  of 
50  miles  of  any  unit  area.  The  radius  of  the  second  zone  is  150  miles,  of 
the  third  zone,  300  miles,  etc.  Postage  rates:  Under  4  ounces,  Ic  per 
ounce;  over  4  ounces,  for  delivery  in  first  zone,  5c  for  first  pound  or 
fraction,  and  3c  for  each  additional  pound. 

For  each  additional  mile  of  delivery  there  is  an  increase  in  the  rate. 
Special  stamps  must  be  used.  Limit  of  weight,  11  pounds.  Consult  your 
postmaster  as  to  rates  and  size  limitations. 

Special  Rates.  For  registering  for  safe  delivery,  10c  in  addition  to 
regular  postage.  Indemnity  limit,  $50.  For  "special  delivery"  or  im- 
mediate dehvery  within  the  carrier-delivery  limit  of  city  free  delivery  and 
within  one  mile  of  any  other  United  States  post  oflice,  10c  in  addition  to 
postage. 

Notes. 

Matter  that  is  harmful,  libellous  or  threatening,  which  has  to  do  with 
lotteries  or  fraudulent  schemes,  is  unmailable. 

At  least  2c  postage  must  be  prepaid  on  first  class  matter.  All  other 
matter  must  be  prepaid  in  full.  Stamps  are  not  affixed  to  publications 
mailed  by  the  publishers,  which  are  weighed  in  bulk.  Stamps  need  not  be 
affixed  to  third  and  fourth  class  matter  consisting  of  at  least  2000  identical 
packages.  In  these  cases,  a  special  "permit"  is  printed  on  the  wrapper 
to  show  that  postage  has  been  paid. 

Postage  Stamps  and  Postal  Cards. 
The  denominations  of  stamps  are  1,  2,  3,  4,  5,  6,  8,  10,  15  and  50  cents; 
and  1  dollar.     Stamped  envelopes  are  issued  of  different  sizes,  for  de» 
nominations  of  1,  2,  4  and  5  cents.     Postal  cards  are  issued  for  Ic,  single, 


68  BUSINESS  ARITHMETIC. 

and  for  2c,  reply.    Private  post  cards  must  be  within  fixed  limits  as  to  size 
and  must  have  a  Ic  stamp  attached. 

2.    Foreign  Mail  Matter. 
Rates  and  Regulations. 

Letters  and  Sealed  Matter.  5c  for  first  ounce;  3c  for  each  additional 
ounce  or  fraction.     See  Domestic  Rates,  for  certain  coimtries. 

Postal  Cards.     2c,  single;  4c,  double- 

Commerical  Paper.  Ic  for  each  2  ounces  or  fraction;  not  less  than  5c 
per  package. 

Printed  Matter.     Ic  for  each  2  ounces  or  fraction 

Samples  of  Merchandise.  Ic  for  each  2  ounces  or  fraction,  but  at 
least  2c  per  packet. 

Parcels-Post.  12c  per  pound.  There  are  so  many  conditions  as  to 
such  mail  that  it  is  advisable  to  consult  your  postmaster  before  sending 
package.     Conditions  are  not  uniform  for  all  countries. 

(For  other  information  see  United  States  Official  Postal  Guide,  or  a 
booklet  on  Postal  Information,  issued  for  public  distribution.) 

EXERCISE. 

Classify  this  mail  matter.  Determine  rate  and  cost  of  postage.  Des- 
tination, the  United  States  unless  otherwise  stated. 

1.  A  2  oz.  letter  to  London.  9.    9  oz.  of  seeds. 

2.  40H  oz.  sealed  ckculars.  10.     12  oz.  seeds,  registered. 

3.  18  oz.  merchandise  samples      H*    A.  6  lb.  package  to  Bremen. 

toBerlm.  12.     3  lb.   7    oz.  of    merchandise. 

4.  Issue  of  60,000  4-oz.  maga-  (^h-st  zone.) 

zines.  13.    A  Christmas  card,  IH  oz. 

5.  8  souvenir  postals  to  London.    14.     4750  lb.  of  newspapers. 

6.  IH  lb.  of  gunpowder.  15.     9  oz.  of  proof  sheets. 

7.  Mdse.,  3  lb.  9  oz.,  to  Zurich.      16.    A  book  weighing  50  oz. 

8.  83^  oz.  commercial  paper  to  Paris. 

17.  Estimate  the  total  postage  on:  21  letters;  150  circulars,  single  page; 
4  pkg.  merchandise,  respectively  9  oz.,  21  oz.,  16  oz.,  13  oz.;  1  pkg.  seeds,  14 
oz.;  2  registered  letters;  5  special  delivery  letters;  25 postals.  (Pkg.— first 
zone.) 

18.  Estimate  the  publisher's  postage  bill  for  his  December  issue  of 
147,456  copies,  weighing  11  oz.  each. 

19.  Compute  postage  on  the  following,  and  state  the  simplest  stamp 
denominations  to  use:  (a)  On  a  4-oz.  letter,  by  special  delivery,  (h)  On 
a  10-oz.  re^stered  package  of  merchandise.     (Second  zone.) 


POSTAGE. 


59 


(c)  On  a  53^  oz.  sample  of  merchandise.     (Fifth  zone.) 

(d)  On  a  book  weighing  47  oz. 

(e)  On  a  3^  oz.  registered  letter  to  London. 

Most  business  organizations,  sending  out  hundreds  or 
thousands  of  pieces  of  mail  matter  per  day,  keep  accurate 
postal  records,  such  as  the  following : 

Postage  Report. 
For  the  month  ending  January  31,  190-. 


Dr7 

Total 
Bought. 

Departments 

Total 

of 
Mo. 

Balance. 

Hand. 

Pur- 
chase. 

Adrer- 
tising. 

Sales. 

Acct'ng. 

Miscel- 
laneous. 

for  the 
Day. 

1 
2 
3 
4 
5 

145 
115 
90 
200 
120 

20 
90 

25 
? 
? 
? 
? 

? 
? 
? 
? 
? 

9 

4 

8 

6 

16 

16 
90 
52 
10 

82 

7 
5 

12 
9 

13 

25 
40 

82 
75 
40 

Ill 

92 

47 

105 

116 

40 

85 
32 
86 
45 

6 

3 

4 

8 

11 

15 
27 
52 
21 
06 

3 

1 

10 
09 
96 
72 
52 

? 
? 
? 
? 

? 

Note.  At  the  close  of  the  day  the  maiUng  clerk  enters  under  each 
department  its  cost  for  postage.  The  ''Total  Bought"  column  shows 
each  day's  purchases;  "Balance"  shows  the  quantity  from  the  previous 
day.  Together,  they  equal  the  amount  on  hand.  The  subtraction  of 
the  total  for  the  day  determines  the  new  balance. 

Two  forms  follow,  which  are  designed  to  show  even  the 
denomination  of  stamps  used,  and  a  classification  of  the  mail 
matter  (page  60). 


On  hand 

Received 

Total 

Used 

Balance 

Value  of  bal. 


Denominations. 


Ic 


54 
700 

? 
528 

? 

? 


2c 


110 
500 

? 
602 

? 

? 


3c 


4c 


90 
400 

? 
128 

? 

? 


5c 


45 

100 

? 

46 
? 
? 


6c 


150 

? 

51 
? 
? 


8c 

200 

? 

40 

? 


10c 

300 

400 

? 

161 
? 
? 


15c 
152 


50c 

40 

100 

? 

45 

? 

? 


1.00 


The  Postage 
Account 


On  hand 

Rec'd 

Total 

Used 

Bal. 


EXERCISE. 
Complete  the  extension  of  the  above  forms,  and  of  the  form  on  page  60, 


60 


BUSINESS  ARITHMETIC. 


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CHAPTER  XII. 

PAYMENT  FOR  SERVICES. 

INTRODUCTORY   EXERCISE. 

1.  Name  advantages  and  disadvantages  of  being  a  wage-earner. 

2.  What  payment  for  services  does  a  private  householder  make? 

3.  Name  typical  cases  in  which  compensation  is  spoken  of  as  a  wage 
As  a  salary. 

4.  What  name  is  given  the  payment  of  a  lawyer?  Of  a  physician? 
How  do  these  payments  differ  from  wages  or  salary? 

The  modern  organization  of  business  demands  a  careful 
accounting  of  all  sources  of  expense.  Of  these,  the  cost  of 
labor  is  a  leading  item  and  requires  special  attention.  Some 
idea  of  the  sums  paid  for  services  may  be  obtained  from  the 
following  recent  Census  figures  for  a  few  selected  industries: 


No.  of  Estab- 
lishments. 

Wage-Earners. 

Ayerage 
Wage. 

Industry. 

Av.  Num- 
ber. 

Wages. 

Agricultural  implements 

648 
1316 
18227 
4956 
4504 
9423 

605 
624 

47,394 

149,924 

81,284 

60,722 

137,190 

402,914 

242,640 
79,601 

$25,002,650 
69,059,680 
43,179,822 
30,878,229 
57,225,506 

229,869,297 

141,426,506 
26,767,943 

$? 
? 

Bakery  products 

Carriages,  etc '. 

? 
? 
? 

? 

General  iron  and  steel 
works 

? 

Silk,  gilk  goods 

? 

Methods  of  payment  vary,  and  the  calculations  involved  are 
often  so  numerous  and  technical  as  to  rank  this  class  of  com- 
putation as  an  independent  subject.  With  millions  of  wage- 
earners,  it  is  evident  that  the  subject  affects,  directly  or  in- 
directly, our  entire  population. 

61 


62  BUSINESS  ARITHMETIC. 

In  business,  the  recompense  paid,  or  stipulated  to  be  paid, 
to  persons,  at  regular  intervals,  for  services  rendered,  is  called 
salary  or  wage.  In  general,  a  stipend  based  on  long  intervals 
(i.  e.,  year  or  month)  is  called  a  salary.  Recompense  reckoned 
by  brief  intervals  (i.  e.,  week,  day,  or  hour)  is  usually  called  a 
wage.  There  are  many  exceptions  to  this  classification.  In 
some  businesses,  payment  for  distinctively  mental  labor  is 
termed  salary  and  manual  labor  wage.  Thus  a  clerk  in  an 
office  receives  a  salary,  while  a  high-class  mechanic,  working  at 
a  machine,  may  receive  a  far  higher  wage. 

In  manufacturing  establishments  payment  for  services  is 
frequently  based  on  the  quantity  of  work  done.  This  work  is 
called  "  piece  work." 

Salaries. 

Salaries  are  usually  paid  monthly  or  semi-monthly.  The 
rate  is  commonly  expressed  in  dollars  per  year.  A  month  is 
considered  one-twelfth  of  a  year,  but  in  many  cases,  pay  for 
a  fraction  of  a  month  is  based  on  the  exact  number  of  days  in 
the  month.  (How  would  one  day's  earnings  in  February 
compare  with  one  day's  in  January?)  Payments  are  usually 
made  by  check  or  in  cash.  Sometimes  employees  are  credited 
on  the  books  and  may  draw  funds  as  needed.  Usually  a  pay 
roll  is  prepared  and  is  signed  by  each  person  when  he  receives 
payment.     A  pay  roll  is  shown  on  the  opposite  page. 


PAYMENT   FOR  SERVICES. 


63 


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64  BUSINESS  ARITHMETIC. 

EXERCISE. 
Reckon  salaries  for  June,  in  the  following  cases: 

1.  Annual  salaries  of  $1200,  $1600,  $900,  $2500,  $450,  $840. 

2.  Promoted  from  $1200  to  $1300  on  June  20. 

3.  Resigned  from  a  $900  position  to  take  effect  on  June  4. 

4.  Appointed  at  $960  to  take  effect  June  4. 

5.  Rule  up  and  extend  the  March  pay  roll  for  the  general  oflfice  force 
of  a  factory.  Assume  salaries  and  positions.  Use  names  of  pupils  in 
the  class.  Have  the  pupils  sign  the  pay  roll.  See  that  signatures  are 
correct.  Illustrate  promotions,  dismissals,  appointments  and  "salaries 
advanced." 

6.  Prepare  a  similar  pay  roll  for  the  teaching  force  of  a  school. 

7.  Special  topic.  (For  individual  pupils.)  Secure  blank  forms  of  pay 
rolls  that  are  in  use  in  typical  establishments  in  your  neighborhood. 
Submit  a  report  showing  how  the  forms  are  adapted  to  special  needs,  and 
illustrating  any  unusual  entries  or  calculations. 

Wages. 

Wages  are  usually  paid  by  the  week  and  are  quoted  at  a 
week,  day  or  hour  rate.  Some  workers  are  paid  by  the  piece 
system,  the  amount  of  wage  depending  on  the  amount  of  work 
done.  In  other  cases,  a  boniis  of  some  form  is  added  for  extra 
efficiency.  Many  establishments  pay  one  and  one-half,  or 
double  rates,  for  overtime  and  necessary  Sunday  work. 

In  many  small  organizations,  the  pay  rolls  in  use  contain  a 
receipt  clause  and  a  column  for  signature.  In  large  organiza- 
tions, the  pay  roll  is  simply  a  record  sheet.  Since  relatively 
few  workmen  keep  business  bank  accounts,  employees  are  paid 
in  cash,  the  pay  envelopes  being  prepared  beforehand;  or  in 
orders  on  the  treasurer  of  the  organization,  payable  on  pre- 
sentation.    A  more  complicated  pay  roll  is  shown  opposite. 


PAYMENT   FOR   SERVICES. 


65 


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66 


BUSINESS   ARITHMETIC. 


From  the  "time  in  hours"  and  "remarks"  columns  explain  how  each 
person  worked,  and  how  his  wage  was  determined?  {Note.  In  the 
above  case,  Sunday  time  and  overtime  is  paid  at  double  rates.) 

Do  you  notice  anything  unusual  in  the  signatures? 


ORAL    EXERCISE. 

Reckon  wages  for  the  week  ending  March  9,  as  follows: 

1.  36  hr.  at  15c;  42  hr.  at  18c;  48  hr.  at  20c;  47  hr.  at  27c;  39  hr.  at  19c; 
42  hr.  at  30c;  54  hr.  at  33ic;  46  hr.  at  25c. 

2.  Reckon  wages  in  the  following  cases,  basing  calculation  on  a  54  hour 
week,  with  double  pay  for  overtime:  42  hr.  at  15c;  54  hr.  at  30c;  54  hr. 
at  32c;  48  hr.  at  17c;  60  hr.  at  25c;  56  hr.  at  30c;  45  hr.  at  35c. 

EXERCISE. 

Rule  up  and  extend  a  pay  roll  for  the  current  week  for  some  shop, 
factory  or  contracting  firm.  Add  to  the  pay  roll  a  column  to  show  work- 
man's position  or  department.     Follow  directions  in  example  5,  page  ^. 

Schedule  of  Change  Desired. 

By 

Address 


In    1-cent  pieces ^ 

"     5-cent      "      ■ 

"  10-cent      "       • 

"  25-cent      "      • 

"  50-cent      "       • 

"  Silver  Dollars 

1 

1 

2 

14 

14 
40 
60 
75 
50 

In  Notes  or  Certificates- 
Si  

41 

36 

20 

100 

2 

'  5 

io 

20 

50 

100 

Total S 

217 

39 

When  the  pay  envelope  method  is  used  the  pay  clerk  com- 
putes, or  if  he  is  an  expert  he  approximates,  the  quantity  of 
each   coin  denomination  required  to  enable  him  to  *'make- 


PAYMENT   FOR  SERVICES. 


67 


up"  each  employee's  exact  "amount  due."  Sometimes  this 
coin  classification  is  incorporated  in  the  pay  roll.  Often 
a  currency  memorandum  similar  to  that  shown  opposite  is 
prepared,  to  be  presented  at  the  bank  when  the  check  for 
the  total  of  the  pay  roll  is  cashed. 

EXERCISE. 

1.  Prepare  a  denomination  slip  for  this  series  of  payments:  $4.78,  $9.62, 
$11.21,  $11.45,  $12.00,  $12.65,  $13.11,  $14  29,  $14.43,  $15.80. 

2.  Prepare  a  currency  memorandum  for  the  pay  roll  shown  on  page  65. 

3.  Prepare  a  wage  pay  roll  and  currency  slip  for  a  contractor's  force 
for  the  current  week.  The  force  consists  of  1  foreman,  4  drivers,  and  17 
loaders.  One  man  is  discharged  and  two  are  hired  during  the  week;  one 
is  out  sick  for  one  day. 

The  pay  clerk  who  is  obliged  to  handle  a  large  pay  roll 
based  on  fixed  rates,  often  buys  or  computes  wage  tables  as 
aids  to  rapid  work.  These  give  the  wages  due  for  any  number 
of  hours  or  half  hours,  at  given  rates  per  week.  The  original 
computations  are  made  with  absolute  accuracy,  and  are  then 
written  in  the  table  to  the  nearest  cent. 

EXERCISE. 
The  Rankin  M.^j^ufacturing  Company. 


$12.00 

48  hours. 

Hour.     Wage.  Hour 

Wage.  Hour. 

Wage.  Hour.  Wage.  Hour.  Wage.  Hour.    Wage. 

k 

13 

8^ 

2 

13 

16i 

4 

13 

24i 

6 

13 

32i 

8 

13 

m 

10 

13 

1 

25 

9 

2 

25 

17 

4 

25 

25 

6 

25 

33 

8 

25 

41 

10 

25 

1* 

i38 

9^ 

2 

38 

17^ 

4 

38 

25^ 

6 

38 

33^ 

8 

38 

4U 

10 

38 

2 

50 

10 

2 

50 

18 

A 

50 

26 

6 

50 

34 

8 

50 

42 

10 

50 

2^ 

63 

10^ 

2 

63 

18^ 

4 

63 

26^ 

6 

63 

34-^ 

8 

63 

42-^ 

10 

63 

3 

75 

11 

275 

19 

75 

27 

6 

75 

35 

8 

75 

43 

10 

75 

3^ 

88 

IH 

2i88 

19^ 

88 

27-^ 

6 

88 

35i 

8 

88 

43^ 

10 

88 

4 

100 

12 

300 

20 

5 

00 

28 

7 

00 

36 

9 

00 

44 

11 

00 

4^ 

1 

1 

20^ 

1.  Complete  the  above  table.     (The  upper  half  only  is  given.) 

2.  What   error  would  result   from   constructing  the  table  by   com- 
putations based  on  the  approximate  value  for  the  first  half  hour? 


68 


BUSINESS   ARITHMETIC. 


3.  Read  off  a  man's  wage,  at  the  given  rate,  for  37  hours,  21  hr,,  13  hr., 
12 >^  hr.,  19  hr.,  363^  hr.,  50  hr.  (double),  52  hr. 

4.  How  might  the  table  be  used  for  a  wage  rate  of  $6.00  per  week? 
Of  $18.00? 

5.  Construct  a  table  for  a  wage  of  $15.00  and  a  week  of  54  hr. 

6.  Construct  a  48  hour  wage  table  showing  overtime  at  double  rate  up 
to  60  hours. 

Teacher's  Note.     Assign  each  pupil  a  different  rate.     Have  tables 
prepared  and  exchanged  for  testing. 

In  some  factories  and  large  stores  time  clocks  are  used  to 
record  the  arrival  and  departure  of  employees.  In  some 
systems,  the  employee  inserts  his  key  in  the  clock,  thus  auto- 
matically recording  his  number  and  the  time.  In  other 
systems,  the  employee  inserts  a  card  in  the  clock,  the  time 
being  stamped  automatically.  Generally,  the  form  is  used 
to  show  whether  or  not  the  employee  is  on  time — small  frac- 
tions of  hours  being  ignored.  In  other 
organizations,  a  few  moments*  tardiness 
may  cause  the  docking  of  the  employee 
to  the  extent  of  one-half,  or  one  hour. 

EXERCISE. 

1.  Compute  the  cost  of  labor  on  the  piece  of 
work  shown  by  ticket  A.  Reckon  to  the  nearest 
one-fourth  hour.  The  rate  given  is  the  rate  per 
week. 

2.  Compute  the  time  of  eight  men  for  Mon- 
day, and  determine  the  wages  due  at  31  ^c  per 
hour: 


FM>r 

kCT 

OYE 

CRY  EXPENSE 
.0.       ^7S 

WOR 

f  €2e-^ 

DATE 

V27 

1 

DAY 

IN 

OUT 

IN 

OUT 

rOTAl 

s 

A.M 

P.M 

s 

A.M 

P.M 

M 

A.M. 

6A« 

11^1 

P.M. 

2253 

5«2 

T 

A.M. 

6l» 

&*2. 

11^1 

V>J1 

P.M. 

-. 

W 

A.M. 

:^c^ 

wiA 

ed 

P.M. 

T 

A.M. 

P.M. 

F 

A.M. 

P.M. 

p 

//S^ff 

fiOUT                                 ' 

Ticket  A. 


No. 
1 
2 
3 

4 
5 
6 

7 
8 


In. 
6:59 
7:01 
6:58 
6:59 

6:58 
6.56 
6:50 


Out. 
12:01 
12 .02 
11.59 
10:01 

12:00 
12:01 
12:02 


In. 

1:00 

1:01 
12:58 

2:00 
12:59 
12:58 
12:57 
12:55 


Out. 
4:59 
5:03 
5.01 
5:02 
5:03 
3:02 
4:59 
5:04 


PAYMENT   FOR  SERVICES. 


69 


Premiums  or  Bonus  Work.  The  premium  plan  of  labor 
payment  is  intended  to  reduce  the  time  of  production  in 
factory  work,  by  the  payment  of  a  bonus  for  extra  efficiency. 
Usually,  as  a  result  of  hundreds  of  experiments,  a  fixed  average 
time  for  a  certain  operation  is  determined.  The  employee 
is  then  told  that  his  employer  will  share  with  him  any  saving 
in  working  time  as  compared  with  this  average.  For  example, 
the  offer  might  be  "one-half  of  the  wage  saved."  If  the 
average  time  to  make  an  article  is  10  hours,  and  the  wage 
$4.00,  the  employee  earns  20  cents  extra  for  eVery  hour  saved. 
The  table  below  shows  the  effect  of  the  plan. 


Time  Con- 
sumed in 
Hours. 

Wage  per 
Piece. 

Premium. 

Total  Cost  of 
Work. 

Workman's 
Hr.  Rate. 

Workman's 
Day  Rate. 

10 
9 

8 
7 
6 

$4.00 

3.60 

3.20 

2  80 

? 

.20 

.40 

? 

9 

$4.00 
3.80 
3.60 
? 

$.40 

.422 

.45 

? 

? 

$4.00 

4.22 

4.50 

? 

? 

Note.  Often  a  set  number  of  pieces  or  operations  is  required  per  hour, 
and  any  increase  in  quantity  is  paid  for  at  a  special  rate.  Often,  if  the 
wage  is  stated  as  a  price  per  piece,  the  wage  rate  is  increased  for  a  quantity 
over  a  certain  fixed  number. 

EXERCISE. 

1.  Complete  the  above  form. 

2.  Prepare  a  similar  table  showing  one-half  hour  time  reductions  from 
6  hours  to  4  hours,  the  6-hour  standard  rate  being  $3.60. 

3.  A  workman,  whose  wage  is  38c  per  hour,  reduces  his  working  time 
on  a  stove  from  4^  to  3  hours,  earning  a  premium  of  16c  per  hour.  He 
raises  his  wage  for  a  9-hour  day  to  $ — . 

4.  The  standard  number  of  pieces  required  of  a  polisher  per  hour  is  12. 
The  rate  is  5c  each,  or  6c  for  extra  efficiency.  He  averages  13^  per  hour, 
for  an  8-hour  day,  earning  $ — . 

Bills  for  Professional  Service. 
Professional  men,  as  doctors,  and  lawyers,  and  many  inde- 
pendent artisans,  charge  fees  for  their  work,  and  render  hills 
for  services.     As  a  rule  the  computations  are  very  simple. 


70  BUSINESS   ARITHMETIC 


B.  C.  Student  &  Co., 
No.  848  Payne  Ave., 

To  Charles  Rawlingson,  Dr., 
Attorney  at  Law. 


Lawyer's  Bill. 

Cleveland,  Ohio,  Dec.  21,  19—. 


For  professional  services  rendered  in  suit  against  Salton  & 

Bro. 
Court  Costs, 
Stenographer,  26  hr.  at  30c 

Note.  Bills  for  the  service  of  an  agent  are  considered  in  the  chapter 
on  Agency. 

EXERCISE. 

1.  Draw  up  a  lawyer's  bill  for  services.  Fee,  $275;  court  costs,  $30; 
stenographer,  17  hr.  at  32c;  registration  fee,  $3.50. 

2.  Draw  up  a  cabinet  maker's  bill  for  remodelUng  an  antique  side- 
board.    Material,  $1.75;  delivery,  50c;  labor  $ — . 

3.  Draw  up  an  original  bill  for  services. 

4.  From  the  doctor's  memorandum  card  illustrated  on  the  following 
page,  showing  house  treatments  (numerals),  and  office  treatments  (O), 
draw  up  a  bill.  The  rates  are  $2  and  $1  respectively,  for  house  treat- 
ments and  office  visits.     Supply  names. 


PAYMENT   FOR   SERVICES. 


71 


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d 

CHAPTER   XIII. 

BUSINESS  TERMS  AND  ACCOUNTS. 

All  business  is  an  exchange  of  one  kind  of  value  for  anothei 
kind.  The  values  exchanged  may  be  general  property,  funds, 
or  services.  In  this  connection,  the  business  man  employs 
two  terms:  (1)  Debit,  as  an  expression  of  value  received  or 
due;  and  (2)  credit,  an  expression  of  value  delivered  or  owed. 

Illustrations.  (1)  A  dealer  gives  $200  (a  credit)  for  a  horse  (debit). 
(2)  The  dealer  pays  his  clerk  $20  salary.  The  debit,  or  value  received, 
is  services;  the  credit  is  $20  cash.  (3)  He  buys  40  bbl.  of  flour  from  Norton, 
giving  his  note  for  30  days.  The  debit  is  merchandise;^  the  credit  a  note 
payable.     Notice  that  in  every  transaction  the  debits  equal  the  credits. 

Accounts  are  collections  of  debits  or  credits  relating  to  a 
particular  person  or  thing.  Thus  the  Cash  Account  is  a  record 
of  all  cash  received  or  paid  out.  Other  accounts  with  which 
the  average  person  meets  are:  Merchandise,  Expense  and 
Personal. 

If,  in  the  course  of  business,  a  transaction,  or  a  series  of 
transactions,  produce  more  than  they  cost,  a  gain  results; 
if  the  sum  spent  exceeds  the  returns,  a  loss  results.  The 
excess  of  gains  over  losses,  for  an  entire  business,  or  for  a 
certain  series  of  transactions,  is  termed  the  net  gain;  the 
excess  of  losses  over  gains  is  a  net  loss.  Any  property  owned 
by,  or  owed  to,  a  person  is  termed  his  resource;  any  amount  he 
owes  is  termed  his  liability.  If  resources  exceed  liabilities, 
a  person  has  a  net  capital,  or  present  worth,  equal  to  the  differ- 
ence. If  liabilities  are  greater,  the  excess  represents  his  net 
insolvency.  Properly  kept  accounts  show  the  resources,  lia- 
bilities, losses  and  gains  of  a  business. 

The  Cash  Account  shows  the  debits  and  credits  of  cash,  the 

72 


BUSINESS   TERMS   AND   ACCOUNTS. 

difference,  or  balance,  representing  the  amount  of  cash 
hand,  as  follows: 

Cash, 


73 


on 


19— 


RECEIVED 


19- 

2000 

— 

€ct. 

18 

Vfi 

— 

20 

485 

60 

24 
31 

2625 

60 

PAID 


*  nee 


30 

m 


2625  60 


Note.  The  halnnce  is  entered  on  the  smaller  side,  so  that  the  sides 
"total"  the  same.  The  source  of  receipt  or  payment  is  sometimes  entered 
in  the  explanatory  space. 

The  Merchandise  Account  shows  the  cost  of  merchandise 
bought  (debits)  and  the  receipts  from  sales  (credits).  In 
general,  explanations  are  not  written.  Here,  however,  they 
show,  among  other  things,  the  people  who  owe,  or  are  owed, 
for  merchandise.  The  balance  shows  a  loss,  if  the  debit  side 
exceeds  the  credit,  or  a  gain,  if  the  credit  side  is  in  excess. 
See  below. 

Merchandise, 


19- 


MDSE.  REC  D. 
^oodi.  on,  Aa/nc6 


5.K>.   So.   ^ane 

i6\ifr^.  ^ou^ 

18,^i^uA, 
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278 

40 

8 

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15 

500 

— 

29 

??? 

— 

31 

2683 

SO 

MDSE.  DELIVERED. 


40 
120 

H 

2146 


50 


60 
20 

30 


Note.  If  unsold  goods  are  on  hand  their  cost  value  (inventory)  is 
entered  on  the  credit  side.  This  is  equivalent  to  subtracting  the  inventory 
from  the  debit  side,  in  order  to  find  the  cost  of  the  goods  sold,  and  thus 
to  determine  the  gain. 

The  Expense  Account,  as  illustrated  on  the  following  page, 
shows  the  cost  of  running  a  business: 


74 


BUSINESS   ARITHMETIC. 
Expense, 


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17 

A  Personal  account  shows  the  amounts  owed  to,  or  by,  the 
person  whose  name  appears  in  the  title. 


Henry  Brown,  1456  K  St.,  N,  W. 


19— 


^Md^. 


^Mc{&e. 


240 
165 

— 

19— 

10 
10 

25 

405 

— 

176 

50 

176 

50 

5 
JfiO 


105 


48 


'i'i 


Note.  Whenever  the  two  sides  of  a  personal  account  are  equal,  it  is 
ruled  up  to  show  no  balance  or  debt.  If  debits  exceed  credits  the  person 
named  owes  the  firm  the  balance;  if  the  credits  are  in  excess  the  firm  owes 
the  person  named. 

The  Proprietor's  or  Partner's  Account  shows  his  relation  to 
the  business.  The  credits  usually  represent  investments 
and  gains;  the  debits  show  the  losses  and  withdrawals.  A 
credit  excess  represents  the  present  worth,  or  net  capital;  a 
debit  excess,  the  net  insolvency.  A  proprietor's  account  is 
shown  on  the  next  page. 


BUSINESS   TERMS   AND   ACCOUNTS. 
S.  p.  Mason,  Proprietor. 


75 


19— 


11 


85 

f 

60 

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12865  SI 

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865 


12865 


ORAL    EXERCISE. 

1.  What  are  debits?     Credits?     WTiat  is  an  account? 

2.  When  is  an  account  debited?     When  credited? 

3.  What  sum  of  cash  was  received,  according  to  the  cash  account? 
What  sum  paid  out?  What  sum  was  on  hand  on  October  21?  For  what 
was  cash  paid  out  during  October? 

4.  From  the  Merchandise  account  it  is  evident  that,  $  ?  worth  of 
merchandise  was  bought  during  January,  and  that,  $  ?  was  sold.  $  ? 
was  sold  for  cash;  $  ?  for  a  note;  and  $  ?  to  be  paid  for  later  by  ?. 

5.  Henry  Brown's  account  shows  that  he  bought  merchandise  on  what 
dates?  For  which  purchases  has  he  paid  in  full?  For  which  has  he 
evidently  paid  in  part? 


6. 


8. 
$2500? 


Name  the  debit  and  the  credit  in  each  of  the  following  transactions: 
(a)  The  fii-m  buys  merchandise  for  cash,  $200. 
(6)  The  firm  sells  A.  B.  Bolton,  merchandise,  $400,  for  which  he 

agrees  to  pay  in  ten  days, 
(c)  The  firm  pays  cash  for  office  furniture,  $120. 
{d)  The  firm  receives  cash  for  some  real  estate.     (Real  Est.  Ac.) 
(e)  The  firm  pays  cash  in  settlement  of  a  gas  bill. 

Classify  the  followdng  as  resources,  liabilities,  losses  and  gains: 
(a)  The  balance  of  the  Cash  Account. 

(6)  A  personal  account  with  debits  of  $500  and  credits  of  $400. 
(c)  Merchandise  account  with  debits,  .$600,  credits,  $700.     No 

goods  on  hand. 
id)  Expense  account  with  debits  of  $46.50  and  credits  of  $13.00. 
(e)  Merchandise  on  hand,  $125. 

What  is  the  present  worth,  if  resources  are  $12,800  and*  liabilities 


9.    What  is  the  net  loss  if  gains  are  $560  and  losses  $764? 


76 


BUSINESS   ARITHMETIC. 


EXERCISE. 

1.  Copy  the  Cash  Account  on  page  73  through  Oct.  24.  Add  these 
entries:  Oct.  25,  received  for  merchandise,  $250;  Oct.  26,  paid  for  horse 
and  wagon,  $185;  Oct.  27,  lost  $2;  Oct.  29,  received  from  C.  B.  Drew, 
$89.60.     Balance  the  account. 

2.  Alter  the  Merchandise  Account  as  follows:  Goods  unsold  on  Jan.  31, 
$2003.15;  amount  paid  by  A.  B.  Brown,  Jan.  15,  $36.40.  Balance  the 
account. 

3.  Find  the  balance  again,  assuming  that  all  goods  had  been  sold. 
What  does  the  account  show? 

4.  Write  an  account  for  D.  Poynter,  Proprietor,  showing  invest- 
ments on  Jan.  1,  of  cash,  $2500;  merchandise,  $3867;  and  real  estate,  $4500. 
Withdrawals,  Jan.  10,  $265  cash;  Jan.  18,  $1645  in  merchandise.  Net 
gain,  Jan.  31,  $211.46. 


NAME 


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FILE  NO. 


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(a)  Brewer  still  owes  $  ? 

(6)  Between  what  dates  did  Brewer  owe  nothing? 

(c)  Between  what  dates  did  he  owe  the  least  amount? 

{d)  How  do  his  cash  payments  compare  with  his  payments  by  note? 

(c)  Explain  the  credit  item  for  Feb.  25. 


CHAPTER  XIV. 

ADVERTISING. 

Modern  advertising,  or  solicitation  through  public  notice, 
really  dates  from  the  insertion  of  such  notices  into  the  maga- 
zines in  1864.  In  that  year,  the  most  prominent  monthly  con- 
tained, in  its  October  number,  three  and  one  quarter  pages. 
During  this  period  advertising  rates  rose  enormously,  as  did 
the  number  and  circulation  of  periodicals.  Advertising  ex- 
tended in  heavy  volume  to  newspapers,  and  then  to  bill  boards, 
complicated  catalogues,  street  car  signs,  show  window  dis- 
plays and  demonstrations.  In  a  recent  year  it  was  estimated 
that  $600,000,000  was  spent  in  these  main  sources  of  adver- 
tising. 

Practically  everyone  reads  advertisements;  and  many  people, 
regularly  or  on  some  special  occasions,  advertise.  The  cost 
of  advertising  is  one  of  the  heaviest  expenses  of  many  a  modern 
business.  The  methods  of  computing  costs  and  returns,  how- 
ever, are  based  on  the  simplest  arithmetical  processes. 

INTRODUCTORY  EXERCISE. 

1.  What  is  advertising? 

2.  Name  at  least  five  ways  of  advertising  merchandise  in  print. 

3.  Name  other  ways  of  advertising. 

4   What  is  a  trade  mark?    Is  it  a  form  of  advertising? 

5.  What  effects  have  climate,  season  and  prosperity  on  advertising 
matter? 

6.  Is  the  paper  of  largest  circulation  necessarily  the  best  advertising 
medium? 

7.  What  is  the  value  of  a  "catch  phrase"  or  title? 

8.  Why  do  not  local  dealers  advertise  in  magazines? 

77 


78  BUSINESS   ARITHMETIC. 

9.  Does  the  character  of  newspaper  advertising  differ  from  that  oi 
magazines? 

10.  Can  you  name  any  article  brought  into  general  use  by  advertising? 

11.  Some  advertisements  give  no  dealer's  name.    Why? 

12.  What  need  has  the  salaried  man  or  private  householder  to  adver- 
tise? 

13.  Special,  Bring  to  class  specimen  advertisements,  securing  cost  of 
insertion,  where  possible. 

Wall  Signs,  for  advertising  purposes,  are  painted  on  blank 
walls  of  buildings,  fences,  bulletins,  etc.  Contracts  are  usually 
for  the  year,  including  painting,  and  exclusive  of  the  vary- 
ing commissions  or  fees  for  rental.  For  bulletins,  the  price  is 
quoted  per  running  foot. 

EXERCISE. 

1.  A  manufacturer  of  breakfast  foods  rents  for  $200,  a  wall  space  of 
40  ft.  by  24  ft.,  and  pays  6c.  per  sq.  ft.  for  the  sign.     Find  the  entire  cost. 

2.  Lawton  &  Co.  rent  spaces  as  follows:  26'  by  20'  for  $250;  20'  by  18' 
for  $4  per  sq.  ft. ;  32'  by  21'  $2.50  per  sq.  ft.  The  charge  for  placing  signs 
is  $4.25  per  sq.  ft.     What  is  the  entire  cost? 

3.  I  contract  for  six  months,  for  a  20'  sign  on  a  standard  bulletin  board 
(10'  high)  at  45  c  per  running  foot  per  month.  What  is  the  cost  of  ad- 
vertising? 

4.  What  is  the  cost  of  a  one-year  contract  for  60  feet  of  bulletin  space 
at  55  c  per  foot  per  month? 

Posters  for  advertising  purposes  are  shown  on  bill  boards. 
These  boards  are  designed  to  accommodate  posters  four  sheets 
high  and  one  to  six  sheets  wide.  Complete  posters  are  spoken 
of  as  4s,  8s,  16s,  etc.,  according  to  the  number  of  sheets. 
The  sheets  measure  28  in.  high  by  42  in.  wide.  A  bill-poster's 
space  is  rented  by  the  month  of  four  weeks,  or  the  year  of 
forty-eight  weeks.  Prices  are  usually  quoted  for  four  weeks' 
display. 

EXERCISE. 

1.  Find  the  dimensions  in  feet  and  inches  of  4,  8,  12  and  16  sheet 
posters. 


ADVERTISING.  79 

2.  Allowing  for  1/10  rene\\:als  of  damaged  sheets,  how  many  sheets  are 
required  for  450  24-sheet  posters? 

3.  To  post  100  16-sheet  posters  for  ten  weeks  cost  a  cereal  company 
16  c  per  sheet  per  month,  or  $ ?, 

4.  Extend  the  following: 

Estimate  of  the  Cost  of  Two  Thousand  16-sheet  Posters. 

Number  required 2000 

For  renewal,  1/5 ? 

For  16  s.  at  $269  per  M $ 

Freight  and  drayage 26.45 

To  be  posted  12  weeks 

560  in  small  towns at  14  c $ 

1200  in  specified  cities at  16  c 

240  in  San  Francisco at  12  c 

$  = 

Street  Car  Posters  are  printed  on  cards,  usually  1 1  in.  by 
21  in.,  though  some  of  other  size  are  used.  The  rates 
average  about  40  c  per  card  per  month,  on  yearly  contracts, 
45  c  on  contracts  for  six  months,  and  50  c  on  three  months 
contracts. 

EXERCISE. 

1.  The  manufacturer  of  a  toilet  soap  contracts  for  3500  cards  at  $47.50 
per  M.,  paying  S18.50  for  the  design,  and  $16.00  for  the  necessary  plates, 
a  gross  cost  of  $ . 

2.  He  distributes  3/5  of  his  supply  (on  six  months  contracts)  throughout 
Middle  West  cities  at  prices  given  above.    This  cost  him  $ ? 

3.  The  balance  he  distributes  in  Pennsylvania  and  New  York  on  a  one 
year  contract,  at  the  above  rate,  except  for  80  for  which  the  price  is  35  c. 
Find  the  cost  of  the  year  contract? 

4.  Prepare  a  statement  showing  the  entire  cost  to  the  manufacturer. 

Newspaper  and  periodical  advertising  absorbs  the  larger 
portion  of  the  money  spent  on  publicity.  Periodical  adver- 
tising is  usually  the  more  costly.  The  common  measure  of 
space  is  the  agate  line,  of  which  fourteen  equal  one  inch  of 
column  space.  One  periodical  charges  $6.00  per  insertion 
per  agate  line,  or  $4000  for  a  full  page  of  9f ''  by  14j". 


80  BUSINESS   ARITHMETIC. 

For  a  standard  magazine  page,  5^"  by  8",  rates  may  run  up 
to  $500  per  insertion. 

Extensive  general  advertisers  usually  engage  agencies  to 
prepare  copy,  select  publications,  and  arrange  for  insertion. 
The  agency  is  paid  either  a  lump  sum  or  a  commission.  In 
some  cases  the  commission  is  paid  by  the  publisher. 

EXERCISE. 

1.  yxom  the  following  table  find  to  two  decimal  places  the  average 
number  of  copies  issued  for  each  member  of  our  population. 

Newspaper  and  Periodical  Circulation.  * 

(Round  numbers). 

Year        Population        Gross  Circulation  Copies  per  capita. 

1850         23,000,000                426,000,000  ? 

1860         31,000,000                928,000,000  ? 

1870         39,000,000             1,509,000,000  ? 

1880         50,000,000             2,068,000,000  ? 

1890         63,000,000             4,681,000,000  ? 

1900         76,000,000             8,168,000,000  ? 

1910         92,000,000            13,105,000,000  ? 

Find  necessary  values  for  the  following : 

2.  The  recent  holiday  numbers  of  ten  standard  magazines  contained 
respectively,  192,  181,  175,  171,  150,  150,  148,  193,  138,  and  127  pages  of 
advertising,  a  total  of pages,  and  an  average  of pages. 

3.  In  the  same  issue,  the  agate  lines  of  advertising  were  43,776,  41,496, 
39,900,  39,047,  34,980,  34,392,  34,088,  33,306,  32,172  and  27,132  Imes. 
Total  lines .    Average  lines . 

4.  Five  great  railway  systems  recently  spent  for  advertising  in  one  year, 
respectively;  $348,457;  $150,647;  $84,335;  $147,564;  $25"l,532— an 
average  of  $ . 

5.  A  half-page  advertisement  in  a  monthly,  $448  per  page  less  one-fifth 
for  time,  costs  a  dealer  $ for  twelve  insertions. 

6.  The  magazine  referred  to  above  contained  168  pages  of  advertising 
in  a  given  number.  Estimate  the  minimum  income.  Why  might  the 
income  be  larger? 

7.  A  novelty  desk  pad  is  advertised  by  its  maker  in  a  20-line  space  for 
3  months,  in  five  monthlies,  having  line  r^ies  as  follows:  $3.50,  $2.75,  $2.50, 
$2.00  and  $1.20.    Compute  the  cost. 


ADVERTISING. 


81 


Note.  Agencies  receive  net  or  gross  rates  from  publishers.  They 
charge  advertisers  the  net  rate,  plus  a  commission ;  or,  if  under  the  gross 
rate  agreement,  charge  the  advertiser  this  rate,  receiving  a  commission  or 
rebate  from  the  publisher. 

8.  Complete  this  statement. 

New  York,  N.  Y.    Nov.  30,  19    . 
Mr.  Brown  C.  Norton 

To  J.  F.  Jamison,  Dr. 
Advertising  Agent. 


Net 

Gross 

The  Johnstone  Weekly 

ip. 

12  times 

5400 

The  World  Magazine 

1  col. 

12  times 

2700 

The  Hour 

ip. 

12  times 

3880 

Every  Moment 

1^  col. 

12  times 

7650 

The  Monthly 

Ip. 

12  times 

3648 

Literary  Magazine 

Ip. 

12  times 

960 

Livelihood 

Icol. 
/lO 

12  times 

1260 

Commission  1 

? 
? 

? 
? 

In  newspaper  advertising,  which  is  largely  local,  many  dealers  make 
contracts  for  a  fixed  amount  of  space  per  annum,  to  be  used  as  required. 
This  space  is  usually  expressed  in  line  measure.  For  "want"  advertising 
and  minor  notices,  rates  vary  with  the  advertising  matter.  The  rate  cards 
of  newspapers  are  often  very  complicated.  Opposite  is  a  small  section  of 
one. 

Extract  from  a  Newspaper  Rate  Card. 

General  Display  Agate  Line. 

Run  of  paper  50c. 

Special  portion,  last  page,  opp.  editorials,  etc.  55c. 
Space  Discounts.     One  Year  Limit. 

5000  Unes 1/20  off.         20000  lines 3/20  ofif. 

10000    "     1/10"  40000    "     1/5     " 

Classification  Rates. 
Agate  Line. 

Amusements 50c  Une.     Public  notices 20c   line. 

Art  sales,  etc 20c     "       Railway  notices 15c     " 

Auctions 20c     " 

Birth,  marriage,  death 

notices $1.00  ea. 

Financial  notices 45c  line. 


Reading  notices,  financial  page,  $2.00. 


Railway  time-tables,  daily  by  the 
month 20c    line. 

Railway  time-tables,  daily  by  the 
year 15c    line. 


82  BUSINESS  ARITHMETIC 

Want  Advertisements. 

Per  Line 1  time     3         7  Per  line 1  time  3         7 

For  sale 18c     25c    40c  Lost  and  Found. .  .8c      20c      35c 

Help  wanted 8       20       35  Positions  wanted.  .5        10       15 

Note.    Eight  words  to  the  line:  five  words  if  set  in  caps. 

ORAL    EXERCISE. 

Use  above  rates  where  none  are  stated  in  problem.     Find  the  cost  of: 

1.  60-line  advertisement,  run  of  paper. 

2.  28-line  advertisement,  general  display. 

3.  5-inch  advertisement,  special  position  on  last  page. 

4.  100-line  advertisement,   special    position  surrounded   by  reading 
matter — the  charge  being  5c.  extra  per  line. 

5.  5000  lines,  contracted  for  55c. 

6.  20,000  lines,  contracted  for  35c. 

7.  10,000  lines,  quoted  rates. 

8.  4-inch  public  notice. 

9.  Three  insertions  of  a  four-line  liOst  and  Found  notice. 

10.  One  month's  insertion  of  a  56-line  railway  time  table. 

11.  A  six-inch  reading  notice  on  the  financial  page. 

12.  Seven  insertions  of  a  six-line  "  position  wanted  "  advertisement. 

13.  A  ten-inch  advertisement  of  a  circus. 

EXERCISE. 

1.  A  12-in.  advertisement  in  a  Chicago  newspaper  @  $4.92  per 
in.  costs  $  ?  per  insertion,  or  for  26  insertions  @  $3.92  per  inch,  per  in- 
sertion, $  ? 

2.  A  Seattle  dealer  contracted  for  12,000  lines  @  49c.  During  January 
he  inserted  advertisements  as  follows :  120  lines,  65  lines,  140  lines,  210  lines, 
560  lines  (twice  running),  28  lines.     What  is  his  bill  for  the  month? 

3.  A  70-line  double  column  advertisement  in  a  Kansas  City  paper,  @ 
$1.40  per  inch,  costs  how  much  for  two  insertions? 

4.  Compute  the  cost  of  this  advertisement  at  quoted  rates. 

WANTED— EXPERIENCED  SALESWOMEN  IN 
books,   stationery,   handkerchiefs,  notions,  neckwear, 
toys,  dolls,  umbrellas,  &c.      Also  bright  young  women 
from  8th  grade  and  high  school. 
Box  462,  Boston,  Mass. 


ADVERTISING.  83 

5.  An  agency  agrees  to  insert  the  above  "want  ad"  in  103  city  papers 
for  $35.00  for  21  words,  and  $11.65  for  each  additional  7  words.  This 
will  cost  $  ? 

Advertisers  watch  the  returns  from  advertising  and  dis- 
card pubHcations  that  fail  to  bring  customers.  Returns  are 
traced  in  many  ways.  For  example,  an  advertiser  may  alter 
his  street  number,  writing  it  "234  State  Street"  in  one  period- 
ical, and  "236  State  Street"  in  another;  or  he  may  offer  a 
different  bargain  in  each  magazine;  or  he  may  insert  in  the 
advertisement  a  reply-slip,  properly  keyed  with  a  number 
or  letter  for  identification.  The  replies  are  then  classified, 
and  the  papers  yielding  the  lowest  cost  per  inquiry,  the  greatest 
number  of  orders,  or  amount  of  sales,  are  rated  as  the  best 
advertising  media. 

EXERCISE. 

1.  The  form  on  p.  84  illustrates  a  cost  per  inquiry  card.  It  is  arranged 
as  a  calendar  showing  the  number  of  inquiries  received  on  each  date. 
Why  are  no  replies  received  on  Jan.  8,  15,  etc.?  Each  horizontal  line  total 
represents  what?  The  vertical  total?  Extend  the  form  and  find  all 
values. 

2.  To  trace  advertisements  month  by  month,  and  to  see  which  of  a 
series  of  advertisements  draws  best,  the  key  in  a  magazine  may  be  changed 
in  each  issue.  For  example,  the  magazines  may  be  numbered  from  10  to  99. 
To  them  is  affixed  a  number  from  1  to  12  to  show  the  issue.  Magazine 
No.  10,  for  a  Februarj^  issue  would  be  represented  by  102,  because  February 
is  the  second  month.  The  form  on  p.  85  is  made  out  for  June,  and  shows 
on  the  left  the  classification  of  calendar  replies.  The  blocks  on  the  right 
(1  to  12)  are  a  classification  of  replies  by  the  advertisements  that  called 
them  forth.     Find  the  totals  and  explain  the  form  in  full. 


84 


BUSINESS   ARITHMETIC. 


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CHAPTER  XV. 
FACTORS  AND  MULTIPLES. 

Thus  far,  attention  has  been  given  to  numbers,  their  expres- 
sion, the  units  of  which  they  are  composed  and  the  funda- 
mental operations  involving  them.  Certain  properties  com- 
mon to  numbers  should  now  be  noted. 

1.  Any  number  may  he  considered  as  the  product  of  two  or  more 
factors.  The  process  of  determining  these  factors  is  termed 
factoring. 

ORAL    EXERCISE. 

1.  Define  a  factor. 

2.  Name  two  factors  of  35,  64,  120,  42,  18,  168. 

3.  2  X  3  X  ?=  48. 

4.  Name  three  factors  of  36,  160,  54,  846. 

5.  2  is  a  factor  of  which  of  these  numbers:  68,  45,  32,  19,  127,  388? 

6.  Name  some  factor  that  is  common  to  48  and  36;  42  and  280. 

7.  Name  two  factors  of  17,  29,  41,  53,  13.  ^Vhat  is  conmion  about  the 
factors  of  these  numbers? 

Classified  according  to  factors,  integral  numbers  are  even 
if  two  is  a  factor,  and  odd  if  two  is  not  a  factor. 

Integral  numbers  are  prime  numbers  if  the  number  itself 
and  one  are  the  only  factors.  A  series  of  numbers  are  mutually 
prime  if  one  is  the  only  common  factor.  All  numbers  not 
prime  are  composite  and  contain  two  or  more  factors. 

ORAL    EXERCISE. 

1.  Name  and  classify  the  numbers  from  1  to  150,  as  even,  odd,  prime 
or  composite. 

2.  Name  four  groups  of  numbers  that  are  mutually  prime. 

3.  May  the  factors  of  a  number  be  composite? 

Note.  There  are  certain  tests  of  divisibility  that  are  useful  in  factor- 
ing and  general  computation.     Thus  a  number  is  divisible  by: 

86 


FACTORS   AND   MULTIPLES. 


87 


2,  if  it  is  even. 

3,  if  the  sum  of  its  digits  Ls  divisible  by  3. 

4,  if  the  number  represented  by  its  two  right  hand  figures  is  divisible 
by  4. 

5,  if  the  right  hand  figure  is  5  or  0. 

6,  if  even,  and  the  sum  of  its  digits  is  divisible  by  3. 

7,  (No  simple  test). 

8,  if  the  number  expressed  by  the  three  figures  to  the  right  is  divisible 
by  8. 

9,  if  the  sum  of  the  digits  is  divisible  by  9. 
10,  if  the  last  figure  is  0. 


EXERCISE. 

1.  Suggest  tests  of  divisibility  by  100,  1000,  10000,  20,  90,  18,  12. 

2.  If  a  number  is  divisible  by  8,  it  is  necessarily  divisible  by  ?,  ?,  or  ? 

3.  If  a  number  is  divisible  by  12,  it  is  necessarily  divisible  by  ?,  ?,  ?,  or  ? 

4.  9,684,324  is  divisible  by  what  factors  under  13? 
Using  divisibility  tests,  determine  at  least  two  factors  of: 

5.  144.        7.     1728.  9.    2240.  11.     5280.         13.    4589. 

6.  320.        8.     19,200.         10.     63,360.       12.     12,345.     14.     12,726. 


EXERCISE. 

Reduce  to  prime  factors: 

1.  7668. 

Being  even,  2  is  a  factor.  The  resulting  factor,  3834,  is 
even,  and  is  therefore  again  divisible  by  2;  the  sum  of  the  digits 
of  1917  (18)  is  divisible  by  3,  a  prime  factor.  The  remaining 
factors  are  found  by  inspection. 

Check  2X2X3X3X3X71=  7668. 

2.  246.  4.     4225.  6.     1728.  8.    2496. 

3.  389.  5.    4128.  7.    63,360.  9.    7884. 
Find  the  prime  factors  common  to: 

10.    378  and  864. 

Since  7  and  16  have  no  common  factor,  division 
need  not  be  carried  further.  It  is  evident  that  the 
factors  common  to  378  and  864,  are  2,  3,  3  and  3. 


11.    984  and  676.        12.     1425  and  5000.        13. 


Solution. 
2     7668 


3834 

1917 

639 

213 

71 


Solution, 
2  378 

864 

3  189 

432 

3   63 

144 

3   21 

48 

7 
672  and  396. 

16 

88  BUSINESS  ARITHMETIC. 

The  product  of  the  common  factors  (in  Ex.  10,  2  X  3  X  3 
X  3)  of  two  or  more  numbers  is  termed  their  highest  common 
factor  (h.c.f.).  The  highest  common  factor  was  once  of 
extreme  importance  in  compHcated  fractional  work,  but  it  is 
now  little  used,  owing  to  the  substitution  of  decimals  for  all 
but  simple  fractions. 

EXERCISE. 

Find  the  highest  common  factor  of: 

1.    678  and  420.  2.     5280  and  320.  3.    546  and  384. 

EXERCISE. 

Factoring  is  often  employed  to  simplify  and  shorten  the  processes  of 
multiplication  and  division.  This  is  done  by  rejecting  common  factors 
from  numbers  that  stand  in  the  relation  of  dividend  and  divisor. 

1.  Why  is  the  quotient  of  5  X  8  divided  by  5  X  4  the  same  as  8  -i-  4? 

2.  Divide  4X7X8  by  3X7X9.  Reject  the  common  factor  and 
divide. 

What  is  the  effect  of  dropping  the  common  factor? 

3.  Divide  12  X  42  X  18  X  6  by  6  X  8  X  9  X  11. 

Solution. 


3  ^ 

*gX  42  X  IS  X^^3  X42  ^  126 

^X-&XS^X11  11  11 

■2- 


11  A 


Indicate  the  products  and  division  as  shown.  Cancel  the  common  factor,  6. 
Cancel  the  factor  9  of  the  divisor  with  the  9  in  18  of  the  dividend,  leaving  2. 
Cancel  the  2  from  dividend  and  from  8  in  the  divisor,  leaving  4.  Cancel  4 
from  the  divisor  and  from  12  in  the  dividend,  leaving  3.  Multiply  the 
remaining  factors. 

4.     14  X  8  X  64  divided  by  11  X  16  X  4  X  7  =  ? 

6.  3X21X84X9-^69X8X6X14  =  ? 
6      52  X  16  X  5  X  11    ^  ^ 

•     13  X  28  X  12  X  15       • 

7.  The  output  of  a  cracker  factory  is  840  doz.  packages  per  hour, 
for  a  nine-hour  day.  This  is  equivalent  to  how  many  cases  of  8  cartons, 
containing  six  packages  each? 


FACTORS  AND   MULTIPLES.  89 

8.  420  cases  of  4  doz.  cans  each,  are  equivalent  to  how  many  packages 
of  each  containing  6  cans? 

9.  At  the  rate  of  $9.60  per  doz.,  what  is  the  cost  of  five  units? 

10.  In  a  given  year  the  coal  consumption  for  heating  an  office  building 
was  1825  tons.  The  price  paid  was  $3.90  per  ton.  What  was  the  average 
daily  cost  for  fuel? 

The  multiple  of  a  number  is  any  integral  number  of  times 
the  number.  A  common  multiple  of  two  or  more  numbers  is 
one  that  is  an  integral  number  of  times  each  of  the  numbers. 
The  least  number  that  is  such  a  multiple,  is  termed  the  least 
common  multiple  (l.c.m.). 

EXERCISE. 

1.  Name  multiples  of  8,  5,  21,  4. 

2.  12  is  a  common  multiple  of  what  smaller  numbers? 

3.  Name  a  common  multiple  of  15  and  25. 

4.  Name  two  common  multiples  of  4  and  6.  What  is  the  least  common 
multiple? 

5.  Is  the  product  of  two  numbers  a  multiple  of  them? 

6.  Is  it  ever  the  least  common  multiple? 

7.  Find  the  1  c.  m.  of  36,  84  and  96. 

To  be  a  common  mutiple  the  number  must  contain  the 
prime-factors  of  each  of  the  given  numbers. 

Illustration. 

Resolve  36,  84  and  96  into  their  prime  Solution. 

factors.     The  multiple  must  have  the  fac-    36  =  2X2X3X3 
tors  "2"  five  times  because  96  contains     84  =  2X2X3X7 
that  number;  the  factor  "7"  once,  because    96  =  2X2X2X2X2X3 
84  contains  it ;  and  two  "  3's  "  because  both 
36  and  96  have  them. 

The  least  common  multiple  is  therefore 
2X2X2X2X2X3X3X7,  or  2016. 

If  the  factors  are  not  determined  by  in- 
spection, arrange  the  numbers  in  a  hori-  2 
zontal  line  and  divide  by  factors  common             2 
to  any  two  numbers.                                              3 

Bring  down  undivided  quotients  that  3  7  8 

are  not  exactly  divisible.    Continue  until 


Solution. 
36        84 

96 

18        42 

48 

9        21 

24 

90  BUSINESS   ARITHMETIC. 

all  quotients  are  mutually  prime.     The  1.  c.  m.  is  equal  to  the  product 
of  the  divisors  and  the  final  quotients. 

1.  c.  m.  =  2  X  2  X  8  (2  X  2X  2)  X  3  X  7  =  2016. 
Find  the  1.  c.  m.  of: 

8.  4,  9,  6,  2.  10.    4,  20,  30,  18.  12.    5,  35,  60,  75. 

9.  9,  12,  15.  11.     4,  2,  8,  32.  13.     3,  6,  5,  92,  18. 

14.  A  steamship  company  has  two  steamers  leaving  New  York,  the 
fii'st  maldng  a  round  trip  in  10  days  and  the  second  in  16  days.  How  often 
do  they  reach  New  York  at  the  same  time? 

15.  A  manufacturing  company  has  on  the  road,  four  drummers,  each 
following  circuits  that  take  respectively  7,  6,  4  and  12  days.  How  often 
can  the  manager  confer  with  the  four  men  together? 

16.  Two  wheels  of  20'  and  16  ft.  circumference  are  geared  together. 
How  often  do  the  wheels  complete  their  revolutions  at  the  same  instant? 

17.  A  novelty  manufacturer  uses  cloth  in  strips  of  3",  4",  9",  6"  and 
12"  in  width.  What  least  width  of  cloth  should  he  buy  in  order  that  he 
may  cut  it  into  any  of  the  given  widths  without  waste? 

FOR  DISCUSSION. 

1.  Find  the  h.  c.  f.  and  1.  c.  m.  of  any  two  numbers.  Compare  the 
product  of  these  values  with  the  product  of  the  numbers.     Explain. 

2.  The  dollar  of  our  currency  is  a  multiple  of  what  other  denomina- 
tion of  coin? 

3.  Name  some  of  our  systems  of  measures  that  are  based  on  a  multiple 
system. 

4.  What  are  the  advantages  of  multiple  over  mutually  prime  measures? 

5.  Show  that  in  determining  the  1.  c.  m.,  all  numbers  that  are  factors 
of  other  given  numbers  may  be  disregarded. 


CHAPTER   XVI. 
REDUCTION   OF  FRACTIONS. 

A  fraction  is  one  or  more  of  the  equal  parts  into  which  a 
whole  may  be  divided.  One  of  the  equal  parts  is  a  fractional 
unit. 

All  fractions  may  be  expressed  as  common  fractions  by  two 
numbers  or  terms  separated  by  a  horizontal  line.  The 
denominator  (lower  term)  names  the  parts  or  size  of  the  frac- 
tional unit;  the  numerator  (upper  term)  shows  the  number  of 
fractional  units  in  the  given  fraction.  In  reading,  the  numera- 
tor is  invariably  read  first. 

^  five        (numerator)         5       „        •  i  ^t 

Illustration.      .  ,  .,     ,j         .     .    \  =  s  =  nve-eightns. 
eighths  (denominator)       8 

INTRODUCTORY    EXERCISE. 

1.  Name  the  unit  in  the  expressions:  15  lb.,  7  da.,  3  wk.,  1/2,  5/6, 
four-ninths,  7/8  yd. 

2.  Which  term  of  the  common  fraction  is  missing  in  the  decimal 
fraction?     How  is  it  expressed? 

3.  Which  is  greater,  1/4  or  1/8?  How  is  the  value  of  a  fraction  afifected 
by  increasing  its  denominator? 

4.  Arrange  in  order  of  value:  1/5,  1/98,  1/72,  1/5000. 
6.     Can  1/3  lb.  and  1/4  A.  be  compared  as  to  value? 

6.  How  many  thirds  is  1?  WTiich  is  greater,  1/3  or  2/3?  How  is 
the  value  of  a  fraction  affected  by  increasing  the  numerator? 

7.  Arrange  m  order  of  value:  3/9,  7/9,  11/9,  5/9. 

It  is  evident  that  any  increase  in  the  numerator  or  decrease 
in  the  denominator  increases  the  value  of  the  fraction,  and 
that  the  value  is  decreased  by  decreasing  the  numerator  or 
increasing  the  denominator.  Moreover,  if  two  fractions  have 
like  numerators  the  larger  fraction  has  the  smaller  denomina- 

91 


92  BUSINESS  ARITHMETIC. 

tor;  if  they  have  like  denominators,  the  larger  fraction  has  the 
larger  numerator. 

Common  fractions  are  proper  fractions  if  the  numerator  is 
less  than  the  denominator;  otherwise  they  are  improper 
fractions. 

Fractions  are  compound,  if  each  term  is  fractional  I  e.  g., 

3;  1.     A  mixed  number  is  an  expression  containing  an  integer 

and  a  fraction  (e.  g.,  6f ). 

The  tendency  in  all  business  and  general  computation  is 
toward  the  use  of  decimals  in  place  of  complicated  or  irregular 
fractions,  so  that  only  the  simple  fractions  are  in  common  use. 
The  business  series  covers  halves,  thirds,  fourths,  fifths,  sixths, 
eighths,  twelfths,  sixteenths  and  twentieths.  To  these  our 
main  attention  will  be  given. 

EXERCISE. 

1.  This  is  one  inch  of  a  ruler  divided  into  eight  parts.  Name  the 
unit  and  the  smallest  fractional  unit. 

2.  In  the  whole  there  are  how  many  halves, 
fourths,  eighths? 

3.  Multiply  numerator  and  denominator  of  1/2 
by  4.  How  is  the  value  affected? 

4.  Divide  numerator  and  denominator  of   6/8 
by  2.  How  is  the  value  affected? 

5.  Why  is  value  unaffected?  {Suggestion.  Note  independent  effect 
of  first  dividing  numerator  and  then  denominator.) 

It  is  evident  that  a  change  of  expression  by  multiplying  or 
dividing  both  terms  of  a  fraction  by  the  same  number,  causes 
no  change  in  value.  A  fraction  is  reduced  to  lower  terms  when 
smaller  numbers  are  found  to  express  the  same  value;  it  is 
reduced  to  higher  terms  when  larger  numbers  are  used. 

ORAL    EXERCISE. 

1.    Reduce  to  thirds:  8,  17,  21,  14,  8,  64. 


"!,f      ; 

■/. '/,  % 

■'A  \  '/, 

1     1 

REDUCTION   OF   FRACTIONS.  93 

2.  Reduce  to  16ths:  3,  4,  12,  9,  21,  5. 

3.  Reduce  to  12ths:  7,  13,  5,  3,  12,  15. 

4.  Reduce  to  20ths:  3/4,  5/4,  7/5,  3/10,  3/2. 
Illustration.    3/4  =  ?/20.    20  -^  4  =  5;  1/4  =  5/20; 

3/4  =  ^-^  =  15/20. 

5.  Reduce  to  16ths:  3/4,  5/8,  5/2,  7/8,  9/4. 

6.  7/6  =  ?/36;  5/8  =  ?/40;  7/3  =  ?/18;  3/7  =  ?/49. 

7.  Is  there  any  limit  to  the  increase  of  size  of  denominator  in  reduction? 

8.  Reduce  to  improper  fractions:  3i,  5^,  16f,  27f. 
Illustration.     In  3  there  are  3  X  4  fourths.     In  31  there  are  12/4 

+  1/4  or  13/4. 

9.  Reduce  to  lOths:  If,  2f,  12 i 

(Suggestion.     Reduce  first  to  improper  fractions.) 
10.    3i  =  ?/12;  If  =  ?/24;  7i  =  ?/20;  9|  =  ?/6. 

EXERCISE. 

1.  Reduce  76  to  a  fraction  with  a  denominator  of  842. 

2.  Reduce  117  to  43ds. 

3.  3/7  =  ?/427;  118/142  =  ?/710. 

4.  Reduce:  16 j^  to  24ths;  192^2^  to  45ths. 

It  is  often  advisable,  in  computations,  to  simplify  a  fraction 
by  reducing  it  to  lower  terms.  In  final  results,  proper  fractions 
are  usually  expressed  in  lowest  terms,  and  improper  fractions 
are  expressed  as  whole  or  mixed  numbers.  It  has  been  noticed 
that  the  value  of  a  fraction  is  not  affected  by  dividing  numera- 
tor and  denominator  by  the  same  number.  Hence  reduction  is 
possible  by  continued  reduction  by  common  factors.  When 
the  reduced  terms  are  mutually  prime,  the  fraction  is  reduced 
to  its  lowest  terms. 

Illustration.     Reduce  585/675  to  its  lowest  terms. 

Analysis  and  Solution.     By  inspection,  5  is  a  common  585  4-  5  _  117 

factor  of  both  terms.  675  4-  5  ~  135 

By  inspection,  9  is  seen  to  be  a  second  common  factor. 

This  really  amounts  to  dividing  the  terms  by  their  117  -^  9  _  13 

greatest  common  di\isor.  135  -i-  9  ~  15 


94  BUSINESS  ARITHMETIC. 

ORAL    EXERCISE. 

1.  Reduce  to  8ths:  3/4,  2,  10/16,  15/24. 

2.  Reduce  to  12ths:  3/4,  5/6,  18/24,  12/36,  40/60. 

3.  Reduce  to  lowest  terms:  15/18  cu.  yd.,  45/69,  720/2400  T. 

4.  R-duce  to  lowest  terms:  160/4,  325/5,  56/8,  120/3. 

5.  Reduce  to  lowest  terms,  or  to  mixed  numbers:  9/2,  15/4,  73/3 
45/12  lb  .  52/8  bu.,  162/16. 

EXERCISE. 

1.  Reduce  to  lowest  terms:  56/124,  128/264,  985/1200,  172/465. 

2.  Reduce  to  whole  or  mixed  numbers:  725/80,  392/56,  1982/1250. 

3.  Simplify:  285/16  T.,  3924/5280  mi.,  453/144  sq.  ft. 


CHAPTER   XVII. 

FRACTIONS— FUNDAMENTAL  PROCESSES. 

Addition  of  Fractions. 

Since  only  like  numbers  can  be  added,  it  follows  that  only 
fractions  or  parts  of  like  units  can  be  added.  Moreover,  since 
the  denominator  of  a  fraction  is  but  a  name  showing  the 
relative  size  of  the  fractional  unit,  only  similar  fractions,  those 
having  the  same  denominator,  can  be  added.  If  fractions 
differ  only  in  denominator,  addition  is  possible  after  reduction 
of  the  fractions  to  a  common  denominator.  In  the  cases 
commonly  arising  in  everyday  usage,  the  least  common 
denominator  is  usually  determined  by  inspection. 

Illustration.     Add  3/5  and  5/6. 

Solution.    By  inspection,  the  common  denominator  is  30. 

3/5  =  18/30. 

5/6  =  25/30. 

3/5  +  5/6  =  18/30  +  25/30  =  43/30,  or  1^. 


ORAL    EXERCISE. 

Which  of  these  groups  can  be  added 

1  without  intermediate  steps? 

Which  cannot  be  added  at  all? 

1.    4/5  +  8/5  +  5/5. 

3. 

5/4  ft.  +  2  lb. 

2.     1/4  +  1/2  bu. 

4. 

1/4  +  1/5  +  1/6. 

5.     1/4  yd.  +  31  yd. 

Find  the  sum  of: 

6.     1/4,  3/4,  2/4,  9  fourths. 

14. 

3/8,  4/5,  7/10. 

7.     1/3  yd.,  2/3  yd.,  5/3  yd.,  3/3  yd. 

15. 

7/12,  2/3,  1/4. 

8.    3/16,  4/16,  7/16, 11  sixteenths. 

16. 

1/2,  2/5,  7/10. 

9.     1/5,  3/5,  7/5,  4/5,  11/5,  9/5. 

17. 

3/20,  4/5,  1/2. 

10.    2/3 +4/5+ 9/15  (=  ?/45). 

18. 

1/3,  3/2,  5/6, 1/12. 

11.     1/3,  1/4. 

19. 

1/3,  2/9,  5/6. 

12.    3/4  ft.,  5/8  ft 

20. 

3/4  ft,,  2/3  ft.,  5/2  ft. 

13.    1/2,  3/4,  5/8,  1/16. 

95 


2. 

3i 

51,  6f. 

4. 

5/12 

,  1^1,  1.5. 

6. 

4H, 

85/r. 

32961 

9.     1292U 

562i 

1828^ 

14729t5i 

13461 

862^ 

4621 

928xV 

18f 

6621 

96  BUSINESS  ARITHMETIC. 


EXERCISE. 

Note.  If  the  numbers  to  be  added  are  mixed  numbers,  the  integers 
and  fractions  rnay  be  added  separately  and  the  two  results  combined. 
This,  however,  is  not  necessary  with  very  simple  numbers. 

1.     U,  U,  5i 

3.     32i,  7f,  8f. 

5.    3i,  41,  12tV. 

7.     2141 

319xV 

4291 

296i 


Note.  In  totalling  lengths  of  cloth,  the  exponents,  1,  2,  3,  are  used  to 
represent  one-fourth,  two-fourths  and  three-fourths  respectively.  Thus 
7621  yd.  are  7621  yd. 

10.  27  pieces  of  percale  are  sold  a  customer  on  an  order.  Find  the 
total  quantity  if  the  respective  piece  lengths  ai"e;  44,  40,  43,  42^,  41^,  40*, 
42S  423,  432^  431^  433^  442^  42,  43S  422,  431^  422^  421,  412^  403,  44S  43S  42', 
43^  42S  43S  403. 

11.  Find  the  cost  of  thirty-one  pieces  of  Scotch  cheviot,  @  37c  per  yd., 
the  pieces  measuring  respectively:  38S  37S  37^,  39,  39^,  40S  42^,  40S  38^, 
391,  40,  382,  391^  40,  41,  392,  383,  392^  40,  401,  402,  39^,  38^,  39S  39^,  40S 
39',  392,  391,  40S  412. 

EXERCISE. 

1.  Find  the  sum  of  any  two  fractions  that  have  the  same  numerator. 
Study  the  solution  and  suggest  a  short  method  for  adding  any  two  such 
fractions. 

Add: 

2.  1/8,  1/3.  4.     1/11,     1/16.  6.    5/8,  5/11. 

3.  1/9,  1/4.  6.     4/9,  4/21.  7.    2/3,  2/9. 

Subtraction  of  Fractions. 

Since  subtraction  and  addition  are  reverse  processes,  the 

principles  of  units  and  common  denominators  apply  as  in 

addition. 

Illustration.     Find  the  difference  between  3/7  and  1/6. 
Solution.     3/7  =  18/42.         1/6  ^  7/42. 

3/7  -  1/6  =  18/42  -  7/42  =  11/42. 


SUBTRACTION   OF  FRACTIONS.  97 

EXERCISE. 

(Solve  mentally  if  possible.) 
Find  the  difference  between: 

1.  1/4  and  1/5.  5.     7/3  and  4/5.  9.    7/8  yd.  and  2/3  yd. 

2.  6/7  and  1/4.  6.     4/3  ft.  and  3/4  ft.     10.     7/40  and  3/20. 

3.  3/3  and  2/9.  7.    7/4  and  4/5.  11.     1/3  and  1/4. 

4.  (1/4  +  1/8)  and  3/4.  8.     7/8  and  7/20.  12.     6/9  and  6/11. 

Note.  If  mixed  numbers  are  given,  whole  numbers  and  fractions  may 
be  subtracted  separately,  except  that  in  case  the  fraction  of  the  subtrahend 
is  greater  than  the  fraction  of  the  minuend,  it  may  be  necessary  to  reduce 
an  integral  unit  of  the  subtrahend  to  fractional  units. 

Find  the  value  of: 

13.  5^  -  3i  18.  5|  -  2.1. 

14.  6|  -  1t\.  19.  388f  -  172r\. 

15.  2f  yd.  -  1|  yd.  20.  32,^^  +  87|  -  23^. 

16.  If  -  5/6  21.  582i  -  96^. 

17.  17/^  -  .8. 

Multiplication  of  Fractions, 
introductory. 

'6/8'  may  be  reaxi  in  the  form  '5  eighths.' 

16  times  5  eighths  are  ?  eighths. 

One-eighth  of  16  is  ?  ;  five-eighths  of  16  are  ? . 

It  is  evident  that  the  product  of  a  fraction  by  a  whole 
number  is  equal  to  the  product  of  the  numerator  and  the 
whole  number,  divided  by  the  denominator.  Thus,  either 
fraction  or  whole  number  may  be  taken  as  multiplier. 

Illustration.     Multiply  5/6  by  124. 
Solutim.    124  X  5/6  =  i?^^  =  ^  =  103i 

EXERCISE. 

(Solve  mentally  if  possible.) 

1.  Multiply  320  by  each  of  these  fractions:  1/4,  1/2,  3/4,  5/8,  4/5, 
3/2,  3/16. 

2.  Find  the  following  fractional  parts  of  48:  1/3,  3/4,  5/6,  7/8,  2/3,  7/6. 
8 


3.    Find  2/3  of  7,  3, 1 

I  18, 

42,  6,  27,  49,  56,  129. 

Multiply: 

4.    425  by  2/5. 

8. 

2/7  by  147.            12. 

5.    25  by  3/4. 

9. 

2/9  by  117.            13. 

6.    37  by  5/8. 

10. 

5/8  bu.  by  726.     14. 

7.    3/8  by  5280. 

11. 

5/6  by  8  X  42.      15. 

98  BUSINES   ARITHMETIC. 


320  rd.  by  9/16. 
142  by  2/3. 
7/12  by  30  X  9. 
364  T.  by  4/3. 

If  one  factor  is  a  mixed  number,  it  may  be  reduced  to  an 
improper  fraction,  although  direct  multipHcation  is  common. 
Illustration.     Multiply  36  by  2^. 

Solution  (1).     36  X  2i  =  36  X  11/5  =  ?i^JJ:  =  395/5  =  791. 
Solution  (2).    36 

7i  (1/5  of  36) 
Z2_ 
79i 

EXERCISE. 

(Solve  mentally  if  possible.) 
Find  the  product  of: 

1.  16  X  U.  5.  165  X  If 

2.  424  X  7f .  6.  3i  X  16  T. 

3.  21  X  ^.  7.  2f  X  $120. 

4.  4i  X  160  ft.  8.  168  X  68t^^. 
Find  the  cost  of: 

9.  7|  yd.  dress  goods  @  $3;  8^  yd.  lining  @  17c;  3/4  yd.  velvet 
@  $3.50. 

10.  420  bu.  corn  @  42Jc;  652  bu.  wheat  @  81|c;  1900  bu.  oats  @ 
32ic. 

Fractions  Multiplied  by  Fractions. 

INTRODUCTORY. 

1/2  in.    =   ?  fourths  of  an  inch  =^   ?  eighths. 
1/4  of  1/2  in.  =  1/4  of  ?  eighths,  or   ?  eighth. 

The  product  of  two  fractions  is  evidently  a  fraction  whose 
numerator  is  the  product  of  the  given  numerators  and  whose 
denominator  is  the  product  of  the  given  denominators.  The 
computation  may  be  simpHfied,  in  some  cases,  by  cancellation. 


MULTIPLICATION   OF   FRACTIONS. 


99 


Illustrations.     Multiply  3/85  by  5/6. 
3X5 


15 


Solution  1. 

3/85  X  5/6  =gg-g=^^^  =  1/34. 

Solution  2. 

By  cancelling  common  factors. 

3/85  X  5/6  =  ^^-^  =  n  X  2  =  i 
17      2 

ORAL    FACILITY    EXERCISE. 

Find  the  product  of . 

1.  1/4  X  1/3.  4.    4/5  X  7/12.  7.     3/8  X  8/3. 

2.  1/2  X  3/4.  5.    5/6  X  6/7.  8.    3/8  X  5/6  X  4/5. 

3.  3/8  X  5/2.  6.    3/2  X  8/9. 


EXERCISE. 

Simplify 

1.  3/4  of  9/24  of  7/15  of  852  bu. 

194. 

2.  3/84  X  5/72  X  ^  X  $8290. 

15 

3.  i^  X  84/36  X  21/22. 

Products  of  Mixed  Numbers. 
With  large  integral  numbers  and  small  fractions,  multiplica- 
tion by  parts  is  common.     In  simple  cases,  the  mixed  numbers 
may   be   reduced   to   improper   fractions   and   the   products 
obtained  as  in  the  previous  case. 


Illustrations. 
Solviion, 


(1)  Multiply  429f  by  127|. 


127J 

i    (1/3X3/5) 

143      (1/3  X  429) 

76i    (127  X  3/5) 

3003  ' 

858     \  (127  X  429) 

429 J 

54702f  Ans. 


100  BUSINESS   ARITHMETIC. 


(2)  Multiply  4i  by  Gf. 

Solution.    4i  : 

=  13/3.     6f 

=  51/8. 

17 

13/3  X  51/8 

13  X-BT      221 

-a-xs    ~   8    ~^^^' 
1 

EXERCISE. 

(Solve 

mentally  if  possible.) 

Multiply: 

1.     U  by  2f. 

7.     4f  by  15i 

2.     2f  by  3/8. 

8.     7f  by  14^. 

3.     If  by  3f 

9.     2341  by  65i 

4.     If  by  li 

10.     3451  by  168i 

5.    4|  by  If. 

11.     1420f  by  17281. 

6.     li  by  2f. 

12.     182f  by  14^. 

Short  Methods. 
Few  fractional  short  methods,  aside  from  cancellation,  have 
a  broad  use.     The  most  valuable  method  is  that  applying  to 
mixed  number  factors  having  the  fraction  "J." 
Case  I.     Integers  alike.     Example.     Multiply  4|  by  4^. 
Analysis  and  Solution. 

By  parts,  4^  X  4|  =  4  X  4  +  4  X  1/2  +  1/2  X  4  +  1/2  X  1/2. 
=  4  fours  +  2  halves  of  4  +  1/4. 
=  5  fours  +1/4 
Condensed  solution. 

4^X4^=5X4  +  1/4=201. 

In  general,  multiply  the  integer  by  the  next  higher  integer  and  add  1/4. 
Case  II.     Integers  unlike.     Example.     Multiply  85  by  11^. 
Analysis  and  Solution. 
By  parts,  8i  by  lU  =  11  X  8  +  11  X  1/2  +  1/2X8  +  1/2  X  1/2- 

=  88  +  1/2  (11  +  8)  +1/4. 

=  88  +  9^  +  1/4  =  97f . 
In  general,  add  to  the  product  of  the  integers  one-half  their  sum,  plus  II4. 

EXERCISE. 


Multiply: 

1.     \2h  by  12i. 

5. 

6^  by  6i. 

9. 

m  by  30i. 

2.    8§  by  8^. 

6. 

61  by  8i. 

10. 

15^  by  5.5. 

3.     15i  by  15i 

7. 

10.5  by  12i. 

11. 

461  by  46i. 

4.    9.5  by  9.5. 

8. 

Uh  by  16i. 

12. 

124^  by  6U. 

DIVISION   OF   FRACTIONS. 


101 


DIVISION  OF  FRACTIONS. 
•    1.  Fractions  by  Integers, 
introductory. 

12  ft.  4-  3  =  how  many  feet? 

12  thirteenths  -r-  3  =  how  many  thirteenths?  =  ?/13. 
-    15/16  -^  5  =  1/5  of  15/16  =  1/5  X  15/16  =  ? 

It  is  possible,  evidently,  to  divide  the  fraction  by  dividing 
the  numerator,  or  multiplying  the  denominator  by  the  integral 
divisor. 

The  reciprocal  of  a  number  is  1  divided  by  that  number. 
Multiplying  the  denominator  by  the  integral  divisor  is  equiva- 
lent to  multiplying  the  fraction  by  the  reciprocal  of  the 
divisor. 


Illustration. 

Divide  18/23  by 

6. 

Solutions. 

3 

48-         3 

23  X-^     23 

1 

(i)i-« 

18 -J- 6      3 
23         23  • 

(2)1-6  = 

(3)l|^6  = 

3 
.^y  1_3 
"23^^     23* 

Divide: 

1.  28/37  by  4. 

2.  5/8  by  6. 

3.  2/3  by  12. 

4.  24/15  by  6. 


EXERCISE. 

(Solve  mentally  if  possible.) 


5.  3/8  by  5. 

6.  21/25  by  7. 

7.  2/3  by  7. 

8.  5/9  by  8. 


9.  5/8  by  15. 

10.  142/56  by  14. 

11.  36/89  by  18. 

12.  22/45  by  33. 


2.     Division  of  Mixed  Numbers  by  Integers. 

Since  the  mixed  number  may  be  reduced  to  an  improper 
fraction,  the  method  does  not  differ  from  that  just  shown.  It 
is  common,  however,  to  divide  directly. 

Illustration.     Divide  54 1  by  6. 


102  BUSINESS   ARITHMETIC. 


'Sobkima.    :(1] 

'-t  =  f. 

491               491      1      491 
9     •  *"        9  ^6"  54   "^^^• 

(2)  6)541 

V 

54 

f  (5/9  - 

r  6  =  5/9  X  1/6  =  5/54) 

EXERCISE. 

(Solve 

mentally  if  possible.) 

Divide: 

1.     If  by  3. 

6. 

16i  by  5.                  9.    382f  by  26. 

2.     12f  by  4. 

6. 

4f  by  3.                   10.     193A  by  15. 

3.     27f  by  9. 

7. 

12t\  by  4.               11.     6721  by  12. 

4.    8f  by  3. 

8. 

2321 1  by  8.              12.     458^^  by  15. 

Fractional  Divisors. 

INTRODUCTORY. 
In  4  there  are  how  many  thirds?    How  many  groups  of  2  thirds? 

4^2=iX?=? 
*       3         2 

Evidently  the  quotient  of  an  integer  by  a  unit  fraction  is 
the  product  of  integer  and  denominator.  If  the  fractional 
divisor  is  not  a  unit  fraction,  its  terms  may  be  inverted  (the 
reciprocal  taken)  and  the  quotient  is  then  the  product  of  the 
dividend  (fractional  or  integral)  and  the  reciprocal  fraction. 

Illustrations.     (1)  Divide  425  by  5/6. 
Solution.    425  -^  5/6  is  equivalent  to  425  X  6/5. 

85       6 
JiQS'X^=  510. 

1 

(2)  Divide  ^gby|. 

9      3  9      4 

Solution,     rs  -^  r  is  equivalent  to  ,-5  X  o  • 
lb     4  lb     o 

3 
^^-3-    4- 


DIVISION   OF  FRACTIONS.  103 

EXERCISE. 

(Solve  mentally  if  possible.) 
Divide: 

1.  1/2  by  2/3.  4.     1/6  by  1/7.  7.     5/8  by  7/12. 

2.  2/3  by  1/5.  5.    2/5  by  5/6.  8.    3/8  by  9/16. 

3.  3/4  by  4/5.  6.     3/7  by  4/11. 

Note.  Simplify  the  process  in  the  following  by  reducing  to  a  common 
denominator. 

9.    3/4  by  7/8.  12.    426  by  3/4.  15.    3/15  by  5/36. 

10.  2/3  by  5/6.  13.     129  by  2/3.  16.     5/8  by  17/140. 

11.  5/12  by  1/3.  14.     3545  by  5/18.  17.  256  by  19/24. 

Mixed  Number  Divisors. 

Since  mixed  numbers  may  be  reduced  to  improper  fractions, 
this  case  reduces  to  that  last  given.  If  the  fractions  are  simple 
it  is  often  advisable  to  reduce  to  a  common  divisor. 

EXERCISE. 
(Solve  mentally  if  possible.) 
Find  the  quotient  of: 

1.  U  -^  U.  3.     U  -^  U.  5.    5/2  -^  1|. 

2.  2  ^  If  4.     2|  ^  31.  6.     51  4-  4|. 

Divide : 

7.  5461  by  li  11.  416f  by  12^ 

8.  672H  by  14^  12.  6546^  by  2f. 

9.  1291  by  66|.  13.  599^  by  7f. 
10.     5821  by  13f .  14.  59^^  by  If. 

Conversion  of  Fractions. 

Owing  to  the  growing  use  of  the  decimal  fraction,  it  is 
necessary  to  have  facility  in  expressing  fractions  in  either 
common  or  decimal  form.  It  is  evident  that  the  denominator 
of  a  decimal  fraction,  though  not  expressed,  is  always  deter- 
mined by  the  decimal  point. 


104  BUSINESS  ARITHMETIC. 

Illustration.     Express  .0825  as  a  common  fraction. 

Solution.     .0825  =  825  ten-thousandths  =  -|^  =  ^. 

lUOUU       400 

EXERCISE. 
Express  as  common  fractions  reduced  to  lowest  terms: 

1.  .125.  5.     18.  9.     .096.  13.  .19275. 

2.  .25.  6.     1.54.  10.     .392.  14.  .088. 

3.  .331.  7.    7.45.  11.     .875.  15.  62.045. 

4.  .0084.  8.    375.  12.     .0625.  16.  1.0458. 

The  reverse  process  of  converting  common  fractions  to 
decimal  fractions  is  far  more  common  than  that  just  given. 
It  is  performed  by  fractional  reduction,  or,  more  commonly, 
by  executing  the  represented  division. 

Illustration.     Reduce  28/85  to  the  nearest  decimal  of  three  places. 
Sohdion.     Carry  out  the  represented  division. 

.329     Ans. 


85)28.00000 

25  5 
'2  50 

170 

800 

765 
35 

EXERCISE. 

(Solve  mentally  if  possible.) 

1.  Convert   into   decimal    form:    1/2,  1/3,  1/4,  1/5,  1/6,  1/7,  (.14|), 
1/8,  1/9,  1/10,  1/11,  1/12,  1/15,  1/16. 

2.  Reduce  to  decimals:  3/4,  2/3,  5/8,  3/8,  1/30,  5/6,  2/9,  5/16. 
Note.     Common  measures  are  frequently  expressed  as  decimals  of 

other  measures. 

3.  Express  1  ft.  aa  a  decimal  of  1  yd.;  1  qt.  as  a  decimal  of  1  gal. 

EXERCISE. 
Reduce  to  equivalent  decimals: 

1.    To  two  decimal  places  with  fractional  remainder:  3/12,  5/11,  5/24, 
11/25,  13,  16,  Ui  17/48,  5^,  8,^^. 


DECIMALS  AND   FRACTIONS.  105 

2.  To  the  nearest  four-place  decimal:  47/52,  5/14,  6/28,  9/32,  21/56, 
9/125,  12/250,  13/80,  11/60. 

3.  Express  100'  as  a  decimal  of  a  mile  (5280'). 

4.  Mechanics  often  use  conversion  tables,  giving  fractions  of  inches 
as  decimals  of  a  foot.  Complete  the  following  table  for  each  1/8"  up  to 
one  foot,  computing  to  four  decimal  places. 

Inch.  Fraction  of  Foot.  Decimal. 

1/8  1/96  .0104 

1/4  1/48  .0208 

QUESTIONS    FOR    DISCUSSION. 

1.  Why  are  not  7ths  and  9ths  common  business  fractions? 

2.  What  has  led  to  the  general  use  of  the  common  business  fractions? 

3.  Are  our  common  measures  adapted  to  the  business  fractions? 

4.  What  fractions  are  used  in  measuring  cloth?  Give  similar  illus- 
trations of  the  use  of  particular  fractions  in  specific  trades  or  professions. 

5.  Why  are  decimals  used  in  place  of  fractions,  in  statistical  tables? 

6.  Name  some  fractions  that  are  simpler,  for  business  use,  than  their 
equivalent  decimals? 

7.  What  is  the  shortest  method  of  comparing  the  relative  value  of 
several  irregular  fractions? 

8.  Why  is  the  value  of  a  fraction  altered  by  adding  the  same  amount 
to  both  numerator  and  denominator? 

9.  What  is  true  of  the  product  of  two  reciprocal  fractions?    Why? 


CHAPTER  XXII. 

FRACTIONAL  RELATIONS  OF  NUMBERS. 

It  is  often  necessary  to  determine  the  fractional  relations 
of  numbers.     There  are  two  common  cases: 

I.  To  Find  What  Part  One  Number  Is  of  Another. 

INTRODUCTORY. 

8  is  what  part  of  12?  Reduce  to  lowest  terms.  Which  number  repre- 
sents the  part?  Which  number  represents  the 
whole  to  which  the  part  is  referred? 

2  sevenths  is  what  part  of  4-sevenths? 

3/4  in.  =  ?/8  in.  3/4  m.  =  what  part  of  7/8 
m.? 


INCH 


Vs  V4  %  h  Vs  %  v& 

I  I  I  I  I 


-v^ 


7/8=-  SMALLEST  UNITS 


3/4= -SMALLEST UNITS        H  [^  cvidcnt  thatthc  part  that  one 

number  is  of  another  is  expressed  as  a 

fraction    having   for   a   numerator    the 

number  expressing  the  part,  and  for  the 

denominator  that  whole  to  which  it  is  referred.    This  is  equally 

true  of  integral,  fractional,   decimal,   abstract    or    concrete 

numbers. 

Illustration.     Example.     I  buy  3/8  interest  in  a  business  from  one 
who  owns  4/5  interest  in  it.     What  part  of  his  holdings  do  I  buy? 

SoMion.     ^^^^^  II  =  relation. 

1^  =  3/8  -^  4/5 = 3/8  X  5/4  =  15/32  the  part  bought. 

4/5 


EXERCISE. 

(Solve  mentally  if  possible.) 
What  part  of: 

1.  14  is  22?  3.     1§  yd.  is  7^  yd.?     5.     14  lb.  is  6^  lb.? 

2.  11  is  15?  4.     1/2  is  1/3?  6.    7  bu.  is  3i  bu.? 

106 


FRACTIONAL   RELATIONS   OF   NUMBERS.        107 

7.  $2.50  is  70  c?  9.     4/11  is  13/16?         11.     5.01  is  1.29? 

8.  3/4  is  5/8?  10.     $326.50  is  $420?    12.     1728  is  1264? 

13.  The  stands  at  a  ball  park  have  a  seating  capacity  of  13,500,  of 
which  3200,  or  ?/?,  are  grand  stand  seats;  1/4  are  bleachers  and  the  balance, 
or  ?  seats,  are  in  the  paviUon. 

14.  In  a  certain  coal  mine  an  investigating  commission  discovered 
that  for  every  ton  of  coal  secured  for  market,  6.45  tons  are  wasted.  What 
part  is  wasted? 

15.  Compare  the  speed  of  two  trains,  one  of  which  covers  four  milea 
in  3  5  minutes,  and  the  other  7  miles  in  6  minutes. 

16.  726  tickets  were  taken  up  at  the  entrance  to  a  hall  seating  1200. 
The  hall  was  approximately  what  part  filled? 

Note.  WTien  an  approximate  relation  is  called  for,  find  the  nearest 
simple  fractional  relation. 

II.    To  Find  the  Whole  When  a  Part  Is  Known 

INTRODUCTORY. 

If  16  ft.  equal  1/5  of  a  quantity,  find  5/5  of  it. 

If  16  ft.  equals  4/5  of  a  distance,  find  1/5  of  the  distance.  Fmd  the  entire 
distance. 

Thus  the  whole  may  be  determined  by  finding  the  value  of 
a  fractional  unit  of  the  given  part,  and  multiplying  this  value 
by  the  number  of  fractional  units  in  the  whole. 

Illustration.     3/8  is  2/3  of  what  number? 
Analysis  and  Solution. 

3/8  =  2  thirds  of  a  number. 

1/2  of  3/8,  or  3/16  =  1  third  of  the  number. 

3  X  3/16,  or  9/16  =  3  thirds  of  the  number,  or  the  number  itself. 
In  brief:  If  3/8  =  2/3  of  a  number,  the  number  =  3/2  of  3/8,  or  9/16. 


Find  missing  values: 

1.  16  =  1/2  of  — . 

2.  35  =  5/6  of  — . 

3.  42  bu.  =  3/7  of  — . 

4.  $5.60  =  4/5  of  — . 

5.  2/3  =  1/2  of  — . 

6.  1/2  -  2/3  of  — . 

7.  750  lb.  =  2/3  of  — . 


EXERCISE. 

8. 

4280  = 

=  1/4  of  — . 

9. 

192  lb 

.  =  4/3  of  — . 

10. 

127  = 

3/4  of  — . 

11. 

2420  lb.  =  3/15  of  - 

12. 

$752  = 

=  3/8  of  — . 

13. 

592  = 

4/13  of  — . 

108  BUSINESS   ARITHMETIC. 

14.  A  railway  speed  of  14  miles  in  IS  minutes  is  equivalent  to  a  speed 
of  how  many  miles  per  hour? 

15.  A  steam  shovel  excavated  47^  cu.  yd.  of  earth  in  4|  minutes. 
This  was  at  the  rate  of  how  many  cubic  yards  per  hoiu"? 

16.  A  yield  of  26  bushels  on  a  trial  plot  of  2/5  of  an  acre  is  equivalent 
to  a  yield  of  how  many  bushels  per  acre? 

17.  A  certain  cloth  shrinks  1/32  in  dyeing.  To  have  930  yd.  of 
finished  product,  one  must  dye  how  many  yards? 

18.  A  man  who  offered  me  $750  for  a  3/8  interest  in  a  certain  patent, 
evidently  valued  the  patent  at  % — . 

19.  A  contractor  has  completed  2/5  of  a  grading  contract  in  48  working 
days.     At  the  same  rate,  how  many  days  work  remain? 

20.  A  manufacturer  replaces  his  old  machines  with  new  ones  of  one- 
fifth  greater  productive  power.  He  runs  on  a  ten  hour  day  basis.  His 
new  machines  should  give  him  his  former  weekly  output  with  a  saving  of 
how  many  hours. 

ORAL   FACILITY   EXERCISE. 

Find  the  single  term  equivalent  to  each  group  of  two  or  more  tei*ms. 
Distinguish  amounts  less  than  zero  by  the  term  minus,  thus  3  —  4  =  —  1. 

1.  Halves  and  thirds.  Two  is  what  part  of  three?  Three  of  two? 
Fmd  1/2  +  1/3,  1/3  +  1/2,  1/3  of  1/2,  1/3  -  1/2,  1/2  -  1/3,  1/2  -  2/3, 
1/3  -^  1/2,  1/2  -^  1/3,  1/3  ^  2/3,  1/2  •-  2/3,  1/3  X  1/2,  1/2  X  2/3,  2/3 
of  1/2,  2/3  divided  by  1/3,  1/2  of  .4,  2/3  of  15,  6  =  1/2  of  ?  ,  12  =  1/3  of  ?  , 
1/2  =  1/3  of  ?  ,  1/3  =  1/2  of  ?  ,  1/3  is  what  part  of  1/2,  1/2  is  what  part 
of  2/3? 

2.  Thirds,  sixths  and  twelfths.  Find  1/3  +  1/6,  2/3  -  5/6,  1/6  -  2/3, 
1/6  ^  1/3,  5/6  -^  5/3,  2/3  -i-  1/6,  2/3  ^  4/6,  1/3  of  1/6,  2/3  X  5/6,  1/3 
+  1/16  +  1/12,  1/6  -^  1/12,  1/3  ^  1/12,  7/12  ^  1/3,  5/12  ^  5/6,  5/6 
-  5/12,  1/3  of  2.4,  5/6  of  72,  7/12  of  18,  4  ==  1/6  of  ?,  1.5  =  5/12  of  ?,  6 
=  1/12  of  1/2  of  ?,  .1/6  is  what  part  of  1/12?  Of  1/3?  Of  5/6?  5/6  is  what 
part  of  2/3?    Of  7/12?     1/6  is  1/12  of  ?    5/12  is  1/3  of  ? 

3.  Halves,  fourths,  eighths  and  sixteenths.  Find  1/2  +  3/4,  3/4  +  5/8, 
7/8  -  U,  9/16  -  5/8,  1/4  -^  3/16;  3/4  of  3/16,  5/8  X  1/2,  5/8  of  3/4, 
1/16  -  3/8,  5/16  ^  3/8,  3/8  of  5/16,  1/4  +  1/8  -4-  1/16,  1/2  times  7/16, 
1/2  -^  9/16,  1/2  X  15/16.  1/2  is  what  part  of  7/8?  Of  15/16?  Of  5/4? 
3/4  is  what  part  of  3/8?  Of  15/16?  Of  If?  What  is  1/2  of  3.6?  Find 
3/4  of  64.  What  is  5/8  of  12?  Find  3/16  of  80.  11/16  of  3.2  =  ?  14 
=  1/2  of  ?  10  =  5/16  of  ?  12  -  3/8  of  ?  12  =  9/16  of  ?  1/2  =  3/4  of  ? 
1/8  =  1/2  of  ?    5/8  =  3/4  of  ?    9/16  =-  1/8  of  ?    9/16  =  3/4  of  ? 


FRACTIONAL   RELATIONS  OF   NUMBERS.         109 

4.  Use  the  answer,  at  each  step,  as  the  basic  value  for  the  next  com- 
putation. 

4  +  12  =  ?  Add  8.  Divide  by  2.  Multiply  by  1/2.  Divide  by  1/3. 
Multiply  by  2/3.  Divide  by  4.  Increase  by  2/3  of  itself.  Multiply  by 
4/5.  Divide  by  2/3.  Multiply  by  3/4.  Change  to  eighths.  What  ia 
the  final  result? 

Teacher's  Note.  Similar  original  exercises  should  be  prepared  by 
the  teacher  and  given  to  the  class,  in  order  to  cultivate  a  high  degree  of 
accuracy  in  handling  the  simple  business  fractions. 


CHAPTER  XXIII. 

ALIQUOT  PARTS. 

Numbers  that  exactly  divide  a  given  number  are  termed 
its  aliquot  parts.  Thus  1, 1^,  1/5,  2,  3,  etc.,  are  aliquot  parts 
of  6. 

ORAL   EXERCISE. 

1.  Name  6  aliquot  parts  of  12.  What  fractional  part  is  each  aliquot 
part? 

2.  Find  1/4  of  200;  of  10.     Find  1/8  of  10,  of  100,  of  1000,  etc. 

3.  What  parts  of  $1.00  are  50c,  10c,  33 ic,  20c,  2c? 

4. .  Which  of  these  numbers  are  ahquot  parts  of  10:  1,  I5,  2,  2^,  3,  3i, 
31,  4? 

5.    Complete  the  following  table: 

Aliquot  Part. 


Number. 

1/2- 

1/3 

1/4 

1/5 

1/6 

1/8 

1/9 

1/10 

1/12 

1/15 

1/16 

1/20 

1. 
10. 
100. 
1000. 

50 

33i 

25 

20 

16! 

12^ 

IH 

10 

8i 

6f 

6i 

5 

1.    Multiplication. 

Aliquot  parts  are  frequently  used  to  simplify  multiplication 
and  division.  The  aliquot  parts  of  10c  and  $1.00  are  espe- 
cially useful  in  business. 

INTRODUCTORY. 

1.  Express  as  aliquot  parts  of  $L00,  25c,  12ic,  2^0,  4c. 

2.  How  does  the  cost  of  several  articles  @  25c  compare  with  their 
cost  @  $1.00? 

3.  The  cost  of  48  yd.  @  25c  =  the  cost  of  48  yd.  at  what  price? 

4.  622  X  33ic  =  522  X  $1/3  =  1/3  of  $  ?  =  $  ? 

110 


ALIQUOT   PARTS.  Ill 

5.  The  cost  of  522  articles  @  $1.00  is  $  ?  The  cost  at  33  fc  is 
what  part  of  the  cost  at  Sl.OCV 

It  is  evident  that  the  cost  of  an  article  at  any  aliquot  price 
of  one  dollar  is  its  simple  fractional  part  of  the  cost  at  one 
dollar. 

Illustration.     Find  the  cost  of  568  yd.  cloth  @  12^c  per  yd. 
Analysis  and  Solution.     12^c  =  $1/8. 

Cost  of  568  yd.  @  12^c  =      Cost  of  568  yd.  @  $1/8 
=  1/8  of  $568  =  $71. 
Note.     Notice  that  in  some  of  the  following  examples  the  aliquot 
lactor  is  the  figure  representing  the  quantity  rather  than  the  price. 

EXERCISE. 

Find  cost  of: 

1.  840  yds.  wash  silk  @  25c;  at  33ic. 

2.  1620  yds.  plaids  @  $1.25;  at  33i;  at  50c. 

3.  580  yds.  cotton  at  6fc,  at  6^c. 

4.  864  yds.  prints  at  12§c;  16ic,  20c. 

5.  164  yds.  Iming  at  12|c,  10c,  8ic 

6.  422  yds   percale  at  25c,  20c,  16|c,  12^0. 

7.  360  doz   eggs  at  12^c,  at  25c. 

8.  560  lb.  lard  at  10c;  8ic,  12^c. 

9.  48  lb.  chocolate  @,  25c,  16fc,  33ic, 

10.  450  lb.  tapioca  @  Q\c,  S^c. 

11.  62  hats  @  $2.50  (aliquot  part  of  $10). 

12.  156  T.  old  iron  @  $16f  ton. 

13.  1520  A.  @  $12.50  per  A. 

14.  1500  blank  books  @  25c. 

15.  196  doz.  buttons  @  16fc. 

16.  124  pr.  shoes  @  $1.25  ($l+25c). 

17.  2500  pineapples  @  12ic. 

18.  1425  posts  @  33ic. 

19.  625  yds.  wire  netting  @  25c  yd. 

20.  364  lbs.  coffee  @  33^0. 

21.  48  lbs.  tea  @  50c;  @  33^0. 

Note.  The  principle  of  interchanging  multiplier  and  multiplicand  is 
frequently  employed.  Thus  the  cost  of  25  yd.  @  88c  is  equivalent  to 
the  cost  of  88  yd.  @  25c. 


112  BUSINESS   ARITHMETIG. 

Compute  the  cost  of: 

22.  33i  yd.   @  78c.  24.    25  bu.  beans  @  $2.60  (260c) 

23.  16f  yd.  lace   @  96c.  25.    50  bu.  clover  seed  @  $4.20. 

ILLUSTRATIVE    EXERCISE. 
100  X  720  =  ? 

100 
25  X  720  =  what  part  of  100  X  720?    25  X  720  =  -y-  X  28  =  ? 

Suggest  the  short  method. 

It  is  evident  that  the  aliquot  method  greatly  simplifies  computation, 
often  substituting  oral  for  written  work. 
Illustration.     Multiply  672  X  33i 

Solution-aliquot.  Solution-regular. 

672  X  33i  =  1/3  of  672  X  100..  672 

100  X  672  =  67200.  33i 

67200  -^  3  =  22400  224 

2016 
2016 
22400 
ORAL    EXERCISE. 

Suggest  short  methods  for  multipHcation  by  12|,  6i,  6f ,  25,  50, 
To  multiply  by  12^,  multiply  by  100  and  divide  by  8. 


1. 

Suggest  she 

►rt  me 

161. 

Illustration. 

Tom 

WTiy? 

Find  the  products  of: 

2. 

25  X  36. 

3. 

33i  X  42. 

4. 

12i  X  184. 

5. 

61  X  960. 

6. 

16!  X  640. 

7. 

20  X  85. 

8. 

33i  X  456. 

9. 

6i  X  64. 

10. 

6!  X  90. 

11. 

50  X  588. 

12. 

25  X  692. 

13. 

25  X  12.4. 

14. 

8i  X  72. 

15. 

3i  X  420. 

16. 

12^  X  8.48. 

17. 

2i  X  880. 

WRITTEN  EXERCISE. 

Find  the  products  of: 

1.  426  X  25  4.    6!  X  968. 

2.  33i  X  2672.  5.     6\  X  884. 

3.  4892  X  m.  6.    25  X  672. 


ALIQUOT   PARTS  113 

7.  2.5  X  876.  12.  6i  X  $6.40. 

8.  250  X  968.  13.  25  X  8.004. 

9.  25  X  48/49.  14.  2^  X  8.4. 

10.  33i  X  66f.  15.  3i  X  .0696. 

11.  6f  X  $1.50.  J6.  333i  X  6.0096. 

MULTIPLES    OF   ALIQUOT    PART. 

The  fraction  representing  an  aliquot  part  has  a  numerator 

of ? 

Find  1/2  of  100;  1/8  of  100;  3/8  of  100;  1/3  of  100;  2/3  of  100; 
1/4  of  100;  3/4  of  100. 

12i  X  248  =  .  37|  =  X  12^. 

S7i  =  what  part  of  100.  37^  X  248  =  what  part  of 
100  X  248? 

Suggest  short  methods  for  multiplying  by  multiples  of 
aliquot  parts. 

ORAL    EXERCISE. 

1.     Multiply  963  by  66i 

Solution.     66f  =  2/3  of  100.     66|  X  963  =  2/3  of  963  X  100  =  32100. 

Note  that  the  fractional  part  of  2/3  may  be  taken  before  or  after  mul- 
tiplication by  100.  Also  one  may  multiply  by  100  and  subtract  1/3  of 
product. 


Multiply: 

2.     124  X  125. 

10.    7i  X  640. 

3.    68  X  75. 

11.    66f  X  12. 

4.    42  X  66|. 

12.     125  X  86. 

5.     16  X  37i. 

13.     168  X  37i 

6.    24  X  66f . 

14.    96  X  112|. 

7.    24  X  62i 

15.     1220  X  150. 

8.     48  X  87i. 

16,     84  X  75. 

9.    54  X  83i 

17.     328  X  37i. 

WRITTEN    EXERCISE. 

1.     75  X  874. 

5.     871  X  920. 

2.    661  X  960. 

6.     83i  X  97.3. 

3.    37^  X  8480. 

7.     71  X  4520. 

4.    62^  X  7244. 
9 

S.    3i  X  872. 

114  BUSINESS   ARITHMETIC. 

9.  125  X  1846.  13.  333iX  942  qt. 

10.  133i  X  276.  14.  137^  X  840  bu. 

11.  1121  X  488.  15.  183i  X  6960. 

12.  233i  X  606.  16.  166f  X  825. 

ILLUSTRATIVE    EXERCISE. 

1.  Find  1/4  of  a  dollar,  3/4  of  it,  2/3,  3/8,  5/8,  4/3,  7/8,  3/40. 

2.  What  part  of  a  dollar  are:  25c,  75c,  66|c,  7^c,  15c,  70c,  $1.25, 
$2.50? 

3.  The  cost  of  240  yd.  @  75c  is  what  part  less  than  the  cost  at  $1.00? 
How  many  times  the  cost  at  25c? 

4.  Suggest  two  short  methods  for  finding  the  cost  at  75c. 

It  is  evident  that  the  cost  at  some  multiple  of  an  aliquot 
price  is  obtained  by  taking  the  aliquot  fractional  part  of  the 
total  cost  at  $1.00. 

Illustration.     Find  the  cost  of  726  yd.  @  66fc. 
Solution.     726  yd.  @  $1.00  cost  $726.     66fc  =  2/3  of  $1.00. 
2/3  of  $726  =  $484,  cost. 

ORAL    EXERCISE. 

1.  Suggest  short  methods  for  finding  the  cost  @  80c,  37ic,  Sl.lOfc. 
83ic,  $1.25. 

Find  the  cost  of: 

2.  290  lbs.  @  40c,  10c,  $1.50. 

3.  42  yds.  @  66|c,  50c,  25c. 

4.  85  yds.  @  80c,  10c,  50c. 
6.     96  lb.  @  62ic,  75c,  83ic. 

6.  24  T.  @  $7.50,  $2.50,  $8.00. 

7.  136  qt.  @  25c,  37ic,  75c. 

8.  48  oz.  @  Sic,  6fc,  66fc,  $1.33i 

9.  120  sq.  yds.  @  7ic,  40c,  75c,  37^c,  66fc. 

10.  90  cu.  yds.  @  $1.33^  $2.66f,  $15,661- 

11.  16  T.  @  $2.50,  $25,  $3.75,  $6.25. 

12.  45  ft.  @  20c,  40c,  $1.20,  $1.60. 

13.  75  yd.  @  $1.24,  $3.36,  $2.40,  $.88. 

14.  12i  T.  @  $4.80,  $5.40,  $6.00,  $12.00. 

15.  37i  yds.  @  64c,  72c,  90c,  24c. 


ALIQUOT   PARTS.  115 

DIVISION. 
INTRODUCTORY    EXERCISE. 

1.  $1.00  will  buy  how  many  pounds  @  25c;  @  33ic;  @  50c? 

2.  $1.00  will  buy  how  many  yd.  of  prints  @  25c?  $240  will  buy  240 
times yd.  or yd. 

3.  Suggest  a  method  of  finding  quantity  when  cost  and  aliquot  price 
are  given. 

4.  37^c  =  what  part  of  a  dollar?  $840  -J-  37ic  =  $840  ^  $?  =  840 
X?  =  — 

5.  Suggest  a  method  of  division  when  the  price  is  a  multiple  aliquot 
price. 

It  is  evident  that  by  substituting  the  aliquot  fraction,  the 
operation  is  redi^ced  to  that  of  division  by  fractions,  and  thus 
to  multipUcation. 

Illustration.     $360  will  buy  articles  @  33ic;  or articles 

@  37|c. 

Solution.  (1)  33ic  =  $1/3.  $360  ^  $1/3  =  360  X  3/1  =  1080,  the  no. 
of  articles  that  can  be  purchased. 

(2)  37^  =  $3/8.     360  -i-  3/8  =  360  X  8/3  =  960,  the  number 
of  articles  that  can  be  purchased. 

ORAL    EXERCISE. 


1. 

State 

rules  for  division  by  375C,  45c,  75c,  66fc, 

$1.16f. 

2. 

Suggest  methods  of  checking  such  divisions. 

Cost. 

Price.         Quantity.                    Cost. 

Price. 

Quantity. 

3. 

$72 

25c                     ?                  9.       $5.20 

$1.25 

4. 

40 

12^c                   ?                10.      48 

.37^ 

5. 

46 

16!c                   ?                11.      60 

1.33i 

6. 

45 

6fc                    ?                12.      54 

1.50 

7. 

21 

8ic                   ?                13.    918 

2.25 

8. 

126 

50c                    ?               14.    468 
WRITTEN    EXERCISE. 

.76 

1. 

$758  will  buy  —  lb.  @  80c. 

2. 

$296  will  buy  —  yd.  @  37ic. 

3. 

$540  will  buy  —  bu.  @  62^ 

4. 

$925  will  buy  —  A  @  $12.50. 

116  BUSINESS   ARITHMETIC. 

5.  $5296  will  buy  —  bu.  @>  75c. 

6.  —  cu.  yd.  ©  62^c  cost  $1600. 

7.  —  oz.  @  7ic  cost  $52. 

8.  —  lb.  @  16f  c  cost  $615.50. 

9.  —  lb.  @  12ic  cost  $6040. 
10.  —  lb.  @  $2.75  cost  $810. 

ILLUSTRATIVE    EXERCISE. 

1.  1.25  is  what  part  of  100.     The  quotient  obtained  by  dividing  by 
25  will  be times  the  quotient  by  dividing  by  100. 

2.  7620  4-  100  = .      12^  = of    100.      7620  -^  12^  = 

times  7620  -h  100. 

3.  Suggest  short  methods  for  division  by  aliquot  parts  of  100,  of  1000. 

4.  37i  is  contained times  in  100. 

5.  Explain:  9640  ^  37|  =  8/3  of  1/100  of  9640  = . 

6.  Suggest  short  methods  for  division  by  62^,  66f ,  7^,  3i,  166|,  125, 
133|. 

ORAL   EXERCISE. 
Find  the  quotient  of: 

1.  2500  ^  20.  7.  1600  -J-  80. 

2.  4200  -r-  25.  8.  920  ^  33i. 

3.  3850  -r-  50.  9.  600  -^  66f. 

4.  7200  -^  33i  10.  1200  -J-  133i. 

5.  240  ^  12i  by  16!.  H.  800  ^  125. 

6.  1200  -5-  75.  12.  2760  ^  150. 

WRITTEN   EXERCISE. 

1.  17800  -^  37i     4.  82984  -^  1.33i.  7.  84020  -^  6i. 

2.  52900  ^  62i.     5.  67500  -^  125.  8.  94368  -5-  116f. 

3.  4890  ^  66i      6.  1284  4-  87^  9.  9218  -^  16i. 

Price  by  the  Hundred  or  Thousand. 

In  the  many  cases  where  articles  are  sold  by  the  hundred  or 
thousand,  computations  are  simplified  by  '^  pointing  off." 

Illustration.     (1)  Compute  the  cost  of  920  lb.  lead  @  $4.00  per  c. 
920  lb.  =  9.2  cwt.     $4.00  X  9.2  =  $36.80  cost. 


ALIQUOT    PARTS.  117 

EXERCISE. 

(Solve  mentally  if  possible.) 

Compute  the  cost  of: 

1.  425  lb.  @  $5.00  per  c.  6.  4580  lb.  @  S5.50  per  c. 

2.  675  lb.  @  $12.50  per  c.  7.  546  lb.  fertilizer  @  85c  per  c. 

3.  1250  yd.  @  $45.00  per  c.  8.  2450  ft.  lumber  @  $33,331  per  M. 

4.  786  ft.  @  $1.25  per  c.  9.  5260  ft.  pine  @  $40  per  M. 

5.  1900  ft.  @  $1.33i  per  c.  10.  4690  ft.  oak  @  $50  per  M. 


CHAPTER  XX. 
PROBLEM  ANALYSIS  AND  SOLUTION. 

In  general,  five  steps  should  mark  the  analysis  and  solution 
of  a  problem. 

1.  A  careful  reading  of  the  problem  to  determine  what  is 
given  and  what  required. 

2.  Analysis,  to  determine  the  relations  between  what  is 
given  and  what  required. 

3.  Selection  of  process  and  of  factors  to  be  used. 

4.  The  computation. 

5.  Checking — of  reasoning  and  of  computation. 

A  problem  should  be  thoroughly  understood  before  its 
solution  is  attempted.  The  process  selected  should  be  the 
simplest  and  most  direct.  The  computation  should  be 
performed  and  set  down  in  the  briefest  and  most  direct 
manner  —  factoring,  cancellation  and  short  methods  being 
used  to  save  time  and  energy.  In  the  written  statement,  care 
should  be  taken  to  use  symbols  accurately,  and  to  exercise 
judgment  in  the  naming  of  important  values. 

There  is  no  uniformity  in  the  method  of  solution  of  problems. 
It  is  essential,  however,  that  the  selected  solution  should 
involve  no  unnecessary  labor,  and  that  the  results  obtained 
by  it  should  be  easily  checked.  All  solutions  of  applied 
arithmetic  problems  require  the  application  of  common  sense 
analysis,  but  this  analysis  is  a  simple  matter,  as  the  following 
illustrations  will  show. 

Illustrations.     (1)  Andrews  offers  a  3/8  acre  plot  of  land  for  $72. 
This  price  is  equivalent  to  a  price  of  how  many  dollars  per  acre? 
Analysis  and  Solution. 

1.    Given  the  price  of  a  part,  to  determine  the  price  of  the  whole. 

118 


PROBLEM   ANALYSIS   AND   SOLUTIONS.  119 

2.  If  3/8  of  an  acre  cost  $72,  8/8  acres  mil  cost  8  times  one-third  of  $72. 

3.  All  given  factors  required  for  solution. 

24 

4.  8  X^^  =  $192.     The  value  of  one  acre  is  $192. 

5.  Check.     3/8  of  $192  =  $72. 

(2)  Working  at  the  rate  of  100  strokes  per  minute,  a  power  pump  dehvers 
iOO  gal.  of  water  per  minute.  How  much  must  the  speed  be  increased 
to  pump  30,000  gal.  per  hour? 

1.  Given  present  speed  and  output  per  minute,  and  required  output 
per  hour,  to  find  speed  increase. 

2.  If  the  output  is  400  gal.  per  minute,  it  is  60  times  400  gal.  per  hour. 
The  speed  must  be  increased  by  the  same  fractional  part  that  the  required 
output  is  greater  than  the  present  output. 

3.  By  multiplication,  find  the  hourly  output.  Compare  the  two  out- 
puts to  find  increase.  By  fractional  multiplication  determine  increase  in 
speed. 

4.  Soluiion. 

60  X  400  gal.  =  24,000  gal.,  capacity  per  hour. 

94  000  ~  4 '     B,equired  capacity  is  5/4  of  present  capacity,  or  1/4  greater. 

1/4  of  100  strokes  =  25  strokes.  Therefore,  the  speed  must  be  in- 
creased 25  strokes  per  minute. 

5.  Check.  An  increase  of  1/4  in  speed  wiU  increase  the  delivery  per 
minute  1/4,  or  from  400  to  500  gal.  A  deUvery  of  500  gal.  per  minute 
=  a  delivery  of  60  X  500  gal.  per  hour,  or  30,000  gal. 

Note.  Numbers  1,2  and  3  are  usually  distinct  mental  processes.  They 
are  not  represented  in  paper  solutions. 

ORAL   EXERCISE. 

1.  32  is  4/7  of  what  number? 

2.  A  pump  that  raises  6400  gal.  per  day  of  8  hours  will  raise gal. 

if  run  10  hours. 

3.  At  84c  per  lb.,  what  is  the  cost  of  3/4  lb.  of  pepper? 

4.  A  factory  is  insured  in  one  company  for  $6000,  and  for  $4000  in  a 
second.  If  damaged  by  fire  to  the  extent  of  $3600,  what  loss  should  each 
company  pay? 

5.  An  improvement  to  a  certain  steam  engine  is  warranted  to  reduce 
the  coal  consumption  2/5.     The  present  consumption  being  15  T  per  day, 


120  BUSINESS  ARITHMETIC. 

this  means  a  saving  of T.,  or  a  saving  in  money,  at  $3.40  per  T, 

of  $ . 

6.  A  train  running  1  mile  in  4/5  of  a  minute  is  running  at  the  rate  of 
miles  per  hour? 

7.  The  report  of  a  cannon  is  heard  20  sec.  after  firing.  How  far 
distant  is  the  cannon  if  sound  carries  65,000  feet  per  minute? 

8.  Divide  10  into  two  parts,  one  being  three  times  the  other. 

EXERCISE. 

1.  A  5  inch  line  is  used  on  paper  to  represent  a  distance  of  60  feet. 
The  paper  length  is  what  part  of  the  true  length? 

2.  By  improvements  in  machinery,  the  output  of  a  factory  is  increased 
3/4.  If,  at  the  same  time,  the  working  time  is  reduced  from  10  to  9  hours, 
what  is  the  net  increase  in  output? 

3.  A  blend  of  tea  consists  of  two  grades,  the  first  forming  3/8  of  the 
mixture.  By  increasing  the  proportion  of  second  grade  1/4,  what  part 
remains  first  grade? 

4.  A  pipe  discharges  32|  gal.  per  sec.  How  many  minutes  will  be 
necessary  to  discharge  10,000  gal.?     What  is  its  discharge  rate  per  hour? 

5.  A  dredge  averages  If  cu.  yard  of  material  per  minute.  At 
that  rate,  it  will  remove  how  many  cu.  yd.  in  a  working  day  of  8  hours. 

6.  How  much  water  must  be  added  to  a  solution  containing  1/10 
ammonia,  to  make  the  solution  1/30? 


CHAPTER   XXI. 
INVOLUTION   AND  EVOLUTION. 

I.     Involution. 

The  products  obtained  by  taking  any  number  two  or  more 
times  as  a  factor  are  termed  powers  of  that  number.  The 
process  of  finding  the  powers  of  numbers  is  termed  involution. 

If  the  number  is  taken  twice,  the  product  is  termed  the 
square  of  the  number;  if  taken  three  times,  the  cube  of  the 
number;  if  taken  more  times,  the  corresponding  power  of  the 
number,  as  the  fourth  power,  fifth  power,  etc. 

Powers  are  expressed  by  writing  the  number  with  a  small 
raised  figure  or  exponent,  showing  the  times  it  is  taken  as  a 
factor. 

Illustrations. 

2  X  2  is  written  2^. 

2X2X2X2is  written  2*. 

Since  the  exponents  represent  the  number  of  times  the  given 
number  is  taken  as  a  factor,  the  product  of  two  powers  of  the 
same  number  may  be  represented  by  writing  the  number  with 
the  sum  of  the  given  exponents  as  the  exponent  of  the  product. 
This  principle  is  of  value,  also,  in  reckoning  high  powers  of 
numbers. 

Illustrations. 

(1)  Represent  the  product  of  2'  and  2*. 
Solution.    2'  =  2  X  2  X  2.  Short  method. 

2*  =  2  X  2  X  2  X  2.  23  X  2<  =  23+<  =  2\ 

23  x2<  =2X2X2X2X2X2X2=  2' 
(2)  Find  the  value  of  3«. 
Solution.    38  =  3*+*  =  3*  X  3*. 

3*  =  3  X  3  X  3  X  3  =  81. 

38  =  3*  X  3*  =  81  X  81  =  6561. 
121 


122  BUSINESS  ARITHMETIC. 

ORAL   EXERCISE. 

Determine  the  value  of: 

1.  3'.  3.    53.  5.    8».  7.    12*. 

2.  2«.  4.    4».  6.    4*.  8.     l(fi. 

9.     Find  the  squares  of  4,  5,  6,  7,  8,  9,  10,  11,  12,  13,  14,  15,  16,  20, 
25,  100,  120,  400,  .2,  .5,  .12,  .006,  1/4,  3/8,  2/3. 

10.  Show  that  a  square  must  end  in  0,  1,  4,  5,  6,  or  9. 

11.  Show  that  a  cube  may  end  in  any  of  the  digits. 

EXERCISE. 

Compute,  by  the  simphfied  method,  the  value  of: 

1.  168.  3.    6.05^  5.     (f)«. 

2.  121*.  4.     .0036«!  6.     (B*. 

11.   Evolution. 

The  process  of  determining  a  number  from  one  of  its  powers 
is  termed  evolution.  If  the  number  is  obtained  from  its  square 
it  is  termed  the  square  root;  if  obtained  from  the  cube  it  is 
termed  the  cube  root.  Roots  of  higher  powers  are  named  horn. 
the  powers,  as  fourth  root,  fifth  root,  etc. 

Illustrations.  The  square  of  6  is  6  X  6,  or  36.  6  is  the  square  root 
of  36. 

The  cube  of  4  is  4  X  4  X  4,  or  64.     4  is  the  cube  root  of  64. 

Roots  are  expressed  with  the  symbol  l/  written  over  the 
number  of  which  the  root  is  to  be  found.  The  power  to  be 
taken  is  shown  by  a  numeral,  except  in  the  case  of  the  square 
root. 

Illustrations.  The  square  root  of  225  is  expressed  V225,  but  the 
square  root  equals  15. 

The  cube  root  of  729  is  expressed  ^729,  but  it  equals  9. 

INTRODUCTORY    EXERCISE. 

1.  Name  at  sight  the  equal  factors  or  square  roots  of  4  (4  =  2  X  2; 
square  root  is  2),  9, 16,  25,  36,  64,  81,  100,  10,000,  121,  144. 

2.  Find  the  squares  of  each  digit.     How  many  places  in  each  case? 


INVOLUTION  AND  EVOLUTION.  123 

3.  Find  the  squares  of  1,  10,  100,  10,000,  .1,  .01,  .001.  tiow  do  the 
numbers  of  places  in  the  squares  compare  with  those  in  the  roots? 

It  is  evident,  from  these  examples,  that  the  number  of 
places  in  any  square  is  twice,  or  one  less  than  twice,  the 
number  of  places  in  the  root.  This  fact  is  important,  as  it  is 
possible,  by  pointing  off  the  square  from  the  decimal  point 
into  periods  of  two  places,  to  determine  the  number  of  places 
in  the  root. 

The  method  of  determining  the  square  root  may  be  under- 
stood, now,  by  studying  the  reverse  process,  using  the  method 
of  multiplication  by  parts. 

Example.     Find  the  square  of  86. 

Solution.  86  =  80  +  6. 

80  +  6  86 

80  +  6  86 

(80  X  6)  +  62  516 

802  ■!■    (go  X  6)  6880 

802  _|_  2(80  X  6)  +  62  =  7396 

It  is  evident  that  the  square  of  a  number,  composed  of  tens 

and  units,  is  equal  to  the  square  of  the  tens  plus  twice  the 

product  of  the  tens  multiplied  by  the  units  plus  the  square  of 

the  units.     It  is  evident,   also,   that  any  number  may  be 

divided  into  two  parts  and  its  square  determined  in  exactly 

the  same  way.     This  composition  of  the  square  is  made  use 

of  in  determining  the  root. 

Illustrations.     (1)  Compute  the  square  root  of  1296. 
Solution. 

12'96(30  +  6  =  36,  root. 
302       =9  00 

3  96(6 
Trial  div.  2  X  30  =    60 

2  X  30  X  6   =360 

62  =    36  3  96 
0 
Analysis. 

Since  there  are  4  places  in  the  square  there  are  1/2  of  4,  or  2  places  in 
the  root.    Point  off  by  2's.  12  represents  the  ten's  place  in  the  root,  and 


124 


BUSINESS   ARITHMETIC. 


96  the  unit's  place.  By  trial,  the  largest  square  in  12  is  the  square  of  3. 
.'.  the  largest  square  of  the  tens  is  30^,  or  900.  Subtracting  the  square  of 
the  tens,  there  are  left  396  units.  Since  the  square  of  the  tens  has  been 
subtracted,  there  is  left  in  the  perfect  square  twice  the  product  of  tens  and 
units  plus  the  square  of  the  units.  As  a  trial  divisor,  use  twice  the  tens, 
or  60.  60  is  contained  in  396  six  times.  Try  6  as  the  required  unit. 
2(30X6)  =360;  6X6  =  36.  The  sum  is  396.  Evidently  6  is  the  required 
unit,  and  36  the  required  root. 

Check.     36  X  36  =  1296. 

(2)  Find  the  square  root  of  1.6384. 


Solution 

1.63'84(1.28  root. 
12  j_ 

2X1.0  =  2.0,  trial,      63(0.3 
2X1.0X  .2  =40 

.22-=  .04       44 
2X1.20  =  2  40,  tFial,  .19  84(0.8 
2  X  1.20  X. 08  =.1920 

.082  =  .0064  19  84 
Check.    1.28  X  08"=  1.6384 


or 


Analysis. 

Point  off  by  2*8,  each  way  from 
the  decimal  point.  If  a  perfect 
square,  the  root  will  contain  one 
whole  number  and  two  decimals. 
1  is  the  nearest  square  of  the  unit 
period  and  the  first  figure  of  the 
root.  Subtract  the  unit  and  bnng 
down  the  first  decimal  group. 
Twice  10  tenths,  or  20  tenths  is  the 
trial  divisor.  The  resulting  quo- 
tient, 3,  is  found  to  be  too  large,  so 
the  next  lower  number,  2,  is  taken 
as  the  next  figure  of  the  root. 
Complete  the  square,  considering  1 
as  the  ten,  and  .2  as  the  units. 
Subtract  the  value  of  the  com- 
pleted square  from  .63  leaving  .19 
and  bring  down  final  group.  As  a 
trial  divisor,  use  twice  the  portion 
of  the  root  already  found,  or  2.40, 
obtaining  the  final  figure,  .08. 
Complete  the  square. 


Thus,  to  determine  the  square  root: 

1. — Beginning  at  the  decimal  point  separate  the  number, 
each  way,  into  groups  of  two  figures.  (The  outermost  period 
on  the  left  may  have  only  one  place.) 

2. — Find  the  greatest  square  of  the  left  hand  period.  Its 
root  is  the  first  figure  in  the  required  root 


Treat  as  whole  number. 

1.63'84(1.28  root, 

1«  =  1 

1 

Trial,  2X10 

63 

2X10X2=40 

22=  4 

44 

Trial  2X120 

19  84 

2X120X8  =  1920 

82=     64 

19  84 

0 

INVOLUTION   AND   EVOLUTION.  125 

3. — Subtract  the  square  from  its  period  and  bring  down  the 
next  period.  As  a  trial  divisor  use  twice  the  root  value  already 
found,  annexing  a  cypher.  Considering  the  quotient  as  the 
new  unit,  complete  the  square  by  taking  twice  the  product  of 
the  tens  multiplied  by  the  units  plus  the  square  of  the  units. 

4. — Subtract  this  value  from  the  last  remainder;  bring  down 
the  next  group,  and  proceed  as  before. 

Noi'E.  (1)  In  the  case  of  fractions,  either  take  the  square  root  of 
numerator  and  denominator  separately,  or  reduce  the  fraction  to  the 
decimal  form. 

(2)  When  the  remainder,  at  any  stage,  will  not  contain  the  trial  divisor, 
write  a  cypher  in  the  root  and  bring  down  the  next  group.  If  there  are 
no  further  periods,  periods  of  two  cyphers  may  be  added  to  the  original 
number  and  the  root  determined  to  any  desired  number  of  decimal  places. 

Since  the  product  of  a  second  power  by  itself  is  a  fourth 
power,  it  is  evident  that  the  square  root  of  a  fourth  power  is 
the  square  of  the  number.  If  the  square  root  of  this  square 
is  then  taken,  the  result  will  be  the  fourth  root  of  the  number. 

EXERCISE. 

Find  the  square  root  of: 

1.  256  5.     1025.9209  9.    29,506,624- 

2.  7225.         6.    52.9984.  10.     329,968,821  (2  dec.  pi.). 

3.  7744.        7.     134,944.81.  11.     .008,907,386,72  (5dec.pl.). 

4.  16,641.      8.     .000,000,494,209.     12.     56,847.02938176  (4  dec.  pi.). 

m  2209  17,161  85,264 

7569  3969  23,104  1,580,049 

17.  Reduce  the  following  to  decimals  and  take  the  approximate  square 
root  to  the  thu-d  decunal  place:  (a)  5/14;  (6)  7/88;  (c)  1/21. 

By  continued  square  root,  find: 

18.  The  fourth  root  of  10,098,  039,  121. 

19.  The  fourth  root  of  3.8416. 

CUBE  ROOT. 
INTRODUCTORY    EXERCISE. 

1.  WTiat  is  the  cube  of  any  number?     What  is  the  cube  root  of  the  cube? 

2.  Find  the  cubes  of  the  digits.  What  is  the  largest  number  of  places 
in  the  results? 


126  BUSINESS   ARITHMETIC. 

3.  Find  the  cubes  of  1,  10,  100,  1000,  .1,  .01.  How  many  places  in 
each  cube? 

It  is  evident  that  the  cube  root  of  a  number  is  one  of  its 
three  equal  factors.  It  is  evident,  also,  that  each  period  of 
three  places  in  the  cube  stands  for  one  place  in  the  root. 

As  in  the  case  of  square  root,  the  method  of  determining 
the  cube  root  may  be  understood  from  a  study  of  the  reverse 
process. 

Illustration.    Find  the  cube  of  46. 
Solution.    Let  46  be  written  40  +  6. 

40  +  6  46 

40+6  46 

(40  X    6)  +  62  276 

40^  +    (40  X    6)  1840 

402  _|.  2(40  X    6)  +  62  2116 

40+6  46 

(402  X   6)  +  2(40  X  62)  +  6'  12696 

408  ^  2(402  X   6)  +    (40  X  62)  84640 
40^  +  3(40*  X  -6)  +  3(40  X  62)  +  6»    =    97336 

From  a  study  of  the  product  it  is  evident  that  the  cube  of 
a  two-place  number  equals  the  cube  of  the  tens  plus  three 
times  the  product  of  the  square  of  the  tens  multiplied  by  the 
units,  plus  three  times  the  product  of  the  tens  multiplied  by  the 
square  of  the  units,  plus  the  cube  of  the  units.  It  is  now  a 
simple  matter  to  reverse  the  process  and  determine  the  cube 
root. 

Illustration.    Find  the  cube  root  of  97,336. 
Solution. 

97'336(46  Separate  into  periods  of  three  places. 

43  64  The  nearest  cube  root  of  the  tens  group 

Trial  4800     33  336  is  4.     Subtract  the  cube  of  4  from  97 

3X402X6=28800  and  bring  down  the  next  group.     Asa 

3X40  X  62  =  4320  trial  divisor  use  three  times  the  square 

6'  =     216     33336  of  the  tens,  obtaining  6  as  a  trial  unit. 

Check.     46X46X46  =  97,336.  The  33,336  units  must  be  made  up  of 

three  times  the  product  of  the  square  of 
the  tens  multiplied  by  the  units,  plus  three  times  the  tens  multiplied  by 
the  square  of  the  units,  plus  the  cube  of  the  units.     These  values  are 


INVOLUTION   AND   EVOLUTION.  127 

determmed  separately  and  totalled  as  shown.  Since  they  total  33,336, 
the  6  is  the  correct  unit  value  of  the  root,  and  46  is  the  cube  root  of  the 
number. 

As  in  the  case  of  square  root,  by  always  considering  the 
figures  of  the  root,  which  have  been  determined  at  any  point 
in  the  solution,  as  tens,  it  is  possible  to  determine  the  next 
figure  of  the  root. 

Illustration.  Find  to  three  decimal  places,  the  cube  root  of 
32.890033664. 

Solution.  32.890'033'664(3.203  +  root. 

3'  27 

Trial,        3X30^  5  890(2 

3X30^X2    =  5400 

3  X  30  X22  =  360 

2»  =  _8              5  768 

Trial,    3X3200^  122  033  664(3 

3X32002X  3       =  92160000 

3X3200  X  32      =  86400 

3»      =  27  92  246  427 

29  787  237  remainder. 

Note.  In  trying  for  the  third  figure  of  the  root,  the  trial  divisor  proves 
larger  than  the  new  dividend.  Therefore  a  cypher  is  written  in  the  root, 
another  period  brought  down,  and  a  new  trial  divisor  used. 

It  is  evident  that  the  steps  to  be  followed  in  obtaining  a 
cube  root  are: 

1. — Beginning  at  the  decimal  point,  separate  the  number, 
each  way,  into  periods  of  three  places  each. 

2. — For  the  first  root  figure,  take  the  cube  root  of  the  great- 
est cube  contained  in  the  left  hand  period.  Subtract  the 
cube,  bringing  down  the  remainder  and  the  next  period  of 
three  places. 

3. — For  succeeding  root  figures,  use  as  a  trial  divisor,  three 
times  the  square  of  the  root  values  already  obtained,  con- 
sidered as  tens.  Use  the  quotient  so  obtained  as  a  trial  root 
value  arid  complete  the  cube  based  upon  it.  If  it  proves 
too  large  reduce  the  unit  value  by  1.  Complete  the  cube, 
subtract  from  the  last  dividend  and  bring  down  the  next 


128  BUSINESS  ARITHMETIC. 

period.  Continue  this  process  until  the  periods  are  exhausted, 
or  the  desired  nlimber  of  decimal  places  in  the  root  have  been 
obtained. 

EXERCISE. 

Find  the  cube  root  of: 

1.  941,192.  6.  4/5,  by  reduction                 729_ 

2.  9.300746727.  to  decimal.  *    9261 

3.  147,197,152.      7.  8.0378843  (4  dec.  12,167 

4.  25  (4  dec.  pi.).  pi.).  10,095,813 

5.  1,083,201,548.8.  1007.65  (5dec.pl.).  12.  .0000000678  (3  dec. 

9.  .001205  (to  3dec.pl.).      pi.). 

GRAPHIC  ILLUSTRATIONS  OF  SQUARES  AND  CUBES. 

Note.     If  deemed  advisable,   this  section  may  be  taken  after  the 
chapter  on  practical  measurements. 

1.    Squaees  and  Square  Root. 

It  has  been  shown  numerically  (page  123)  that  the  square  of 

any  number  composed  of  tens  and  units  is  equal  to  the  square 

of  the  tens  plus  twice  the  product  of  the  tens  multiplied  by  the 

c  X  D    units  plus  the  square  of  the  units. 

This  may  be  shown  graphically,  as 

^  follows: 

Illustration.    Compute  the  square  of 
45. 

Solviwm. 

(1)  (2) 

40+5  45 

40+5  45 

5X40+5«  225 

40^+      5X40+ 180 

40»+2  X5  X40+52  =2025  2025 
(3)  To  any  scale,  lay  off  40,  and  then  5,  in  one  straight  line.  The  square 
on  this  line,  ABCD,  is  evidently  the  square  of  45.  Construct  the  square 
of  40,  AB'C'D',  extending  the  sides  meeting  at  D',  to  their  intersection 
with  the  square  of  45.  This  divides  the  square,  as  shown,  into  a  square  of 
40,  two  rectangles  measuring  40  X  5,  and  a  small  square  of  5.  The  square 
of  45  plainly  contains  the  square  of  the  tens  (40),  the  square  of  the  units 


*-5(m) 

40  (0 

5U)-* 

5 

6(u)- 

/ 

i 

40(0 

6(«)- 

i-^ — 

f.\ 

20 

1 

I 

5x 

100 

20 

INVOLUTION   AND   EVOLUTION.  '         129 

(5)  and  twice  the  product  of  units  and  tens  (2X40X5).  But  45  might 
have  been  any  number  of  t  tens  and  u  units.  Substituting  t  and  u  for  40 
and  5  respectively,  it  is  evident,  from  the  figure,  that  the  square  of  t+u 
=  t^-\-2tu-\-u^. 

In  a  number  containing  more  than  two  places,  as  125,  the  principle 
still  holds.  125  =  100+20+5.  As  shown  by  the  diagram,  the  square  of 
100+20  =  1002+2X100X20+202.  Now  as- 
suming 120  as  twelve  tens,  the  square  of  125 
is  evidently,  1202+2X5X120+52.  It  is  evi- 
dent that  each  square,  from  the  inmost,  be- 
comes in  turn  the  square  of  the  "tens"  for  the 
determining  of  the  square  of  the  "tens  plus 
units." 

In  the  reverse  process  of  obtaining  the 
square  root  of  a  number  (see  page  124)  it 
should  be  evident,  now,  that  in  taking  as  the 
highest  figure  of  the  root  the  square  root,  of  ^"^ 

the  greatest  square  contained  in  the  left  hand 

group  of  two  figures,  one  obtains  the  edge  of  the  innermost  square.  Con- 
sidering this  as  the  "tens"  value,  the  next  place  is  computed  as  the  "unit," 
thus  obtaining  the  next  larger  square.  Then  this  square  is  taken  in  turn 
as  the  square  of  the  "tens"  and  a  new  "tens  plus  units"  square  is  obtained, 
and  so  on. 

In  the  process  of  computing  the  square  root,  at  each  step,  after  the 
square  of  the  known  "tens"  is  subtracted  from  the  total  square,  the 
remainder  equals  2  X  tens  X  units  plus  the  square  of  the  units.  The  new 
unit  is  unknown,  but,  by  dividing  the  remaining  area  by  the  known  factor 
of  the  large  term,  "2  tens,"  as  a  trial  divisor,  there  is  obtained  an  approach 
to  the  exact  imit.     This  imit  is  then  tested  to  see  if  it  is  exact* 

2.     Cubes  and  Cube  Roots. 

It  has  been  shown  numerically  (page  126)  that  the  cube  of 
any  two-place  number  equals  the  cube  of  the  tens  plus  three 
times  the  product  of  the  square  of  the  tens  multiplied  by  the 
units,  plus  three  times  the  product  of  the  tens  multiplied  by 
the  square  of  the  units,  plus  the  cube  of  the  units.  As  in  the 
case  of  squares,  this  may  be  shown  graphically  as  well  as 
numerically. 

Illustration.    Compute  the  cube  of  15. 
10 


130 


BUSINESS   ARITHMETIC. 


ions. 

(1) 

(2) 

15 

10+5 

15 

10+5 

75 

5X10+52 

15 

102+  5X10 

225 

102+2X5X10+52 

15 

10+5 

1125 

5X102+2X52X10+5' 

225 

103+2X5X102+ 

52x10 

3375  =  103+3X5X102+3X52X10+5' 
(3)  To  any  scale,  construct  a  cube  of  one  "ten"  edge.     Extending  the 
edges  meeting  at  one  point,  construct  over  the  cube,  a  second  cube  of  an 
edge  equal  to  one  "  ten  "  and  five  "  units."     Assume  the  faces  of  the  smaller 

cube  extended  to  cut  the  faces 
of  the  larger.  A  study  of  the 
subdivided  figure  will  show 
that  the  entire  cube  is  made 
up  of  a  cube  of  10  edge;  three 
rectangular  solids.  A,  B  and 
C,  each  measuring  10X10X5; 
three  rectangular  solids,  D,  E 
and  F,  measuring  10X5X5; 
and  a  cube,  G,  of  5  edge.  Thus, 
as  in  the  numerical  statement, 
the  cube  contains  the  cube  of 
the  tens,  plus  three  solids  whose 
contents  equal  the  square  of 
the  tens  multiplied  by  the  units 
(10X10X5)  plus  three  solids 
whose  contents  equal  the  pro- 
Original  cube  1,  2,  3,  4,  5,  6,  7,  8  shown  duct  of  the  tens  multipHed  by 
hy  th.Q  light  Wne^.  the  square  of  the  units  (10X5 

X5)  plus  the  cube  of  the  units 
(53) ,  Since  this  is  general  for  two  place  numbers,  or  for  numbers  treated 
as  tens  and  units,  the  cube,  in  general,  will  equal  t^-\r^t'hj,-\rZtu^-\-v?. 

In  reconstructing  this  cube  from  the  known  cube  of  the  tens,  it  is  evident 
that  there  must  be  added  to  the  cube  of  10,  3375  - 1000,  or  2375,  cubic  units. 
These  additions  must  be  in  the  form  of  three  solids  having  an  outer  surface 
area  of  10X10,  or  100  square  units;  three  solids  having  an  outer  surface 
length  of  10  and  for  width  the  unit  value  still  unknown;  and  a  cube  having 
the  unknown  unit  for  an  edge.     All  these  additions  have  a  uniform  thick- 


INVOLUTION   AND   EVOLUTION.  131 

ness  equal  to  the  desired  unit  of  the  root.  Since  the  numerical  value 
of  the  cubic  contents,  when  divided  by  the  numerical  value  of  surface  area, 
gives  a  Unear  measure  as  a  quotient,  the  trial  process  in  finding  the  second 
and  succeeding  figures  in  a  cube  root  consists  in  dividing  the  cubic  contents, 
remaining  after  the  cube  of  the  known  root  is  subtracted  by  the  surface 
area  of  the  new  additions,  so  far  as  known.  The  known  part  is  3t^-\-St. 
Since  this  does  not  represent  the  complete  outer  surface  area  of  the  addi- 
tion, it  is  used  as  a  trial  divisor,  and  the  resulting  quotient  is  tested  to  see 
if  it  is  exact,  or  too  large  or  small.     It  is  tested  of  course  in  the  formula 


CHAPTER  XXII. 

DENOMINATE  NUMBERS. 

Standard  Measures. 

The  modern  systematizing  of  business,  and  the  simplifying 
of  methods  of  handling  merchandise,  have  had  a  marked  effect 
on  our  systems  of  measures.  It  is  rarely,  now,  that  commercial 
quantities  are  expressed  in  several  denominations.  Instead, 
some  measure  of  a  table  is  selected  as  a  unit  or  standard  and 
quantities  are  expressed  as  multiples,  decimals,  or  simple 
fractions  of  that  unit.  The  unit  varies  for  different  trades, 
professions,  or  different  classes  of  merchandise.  Thus  the 
civil  engineer  measures  by  units  of  1000  ft.,  100  ft.,  10  ft., 
tenths  of  a  foot,  etc.;  the  mechanic,  in  inches  and  decimals. 
The  contractor  often  estimates  in  cubic  yards  and  decimals. 
Scales  are  now  made  to  weigh  in  tons  and  decimals,  and  clocks 
to  mark  time  in  hours  and  hundredths.  The  grocer  uses  frac- 
tional pounds  rather  than  ounces,  as  the  dry  goods  dealer  uses 
the  yard  and  its  fractions.  Many  milk  dealers  use  the  pint 
as  a  unit.  Grain,  and  some  vegetables,  may  be  measured  by 
the  pound  unit  instead  of  by  dry  measure.  Practically  all 
forms  of  statistics  use  a  decimal  system  based  on  a  single  unit. 

The  tendency  toward  unit  measures  is  aided  and  reflected 
by  the  increasing  sale  of  merchandise  in  unit  packages,  and 
by  the  increasing  use  of  automatic  weighing  and  computing 
machines,  practically  all  of  which  work  on  a  decimal  basis. 

The  simplicity  in  actual  use  is  reflected,  naturally,  in  com- 
putations. Except  in  computations  of  time  and  English 
money,  it  is  seldom  necessary  to  handle  three  denominations 
of  any  table. 

132  - 


DENOMINATE   NUMBERS. 


133 


Tables  of  Measures. 

This  section  includes  the  standard  tables  in  general  use, 
and  a  few  units  and  special  values  common  to  certain  type 
businesses. 

MEASURES  OF  CAPACITY. 

Dry  Measure. 
2  pints     =  1  quart. 
8  quarts  =  1  peck  (pk.). 
4  pecks    =  1  bushel  (bu.). 
1  dry  qt.  =67.2  cu.  in. 
1  bushel  =  2150.42  cu.  in. 

A  heaped  bushel,  used  for  corn 
in  the  ear,  apples,  etc.,  contains 
2747.71  cu.  in. 


Liquid  Measure. 
4  gills  (gi.)  =  1  pint  (pt). 
2  pints         =  1  quart  (qt.). 
4  quarts       =  1  gallon  (gal.). 

1  gallon       =  231  cu.  in. 

Barrels  and  hogsheads  vary  in  size. 
For  general  estimates. 
3U  gal.        =  1  barrel  (bbl.). 
63  gal.         =  1  hogshead. 

MEASURES 

Avoirdupois  Weight. 
16  ounces  (oz.)  =  1  poimd  Qh ). 
100  pounds  =  1  hundred- 

weight (cwt.). 
2000  pounds  or  20 

hundredweight  =  1  ton  (T,). 
In   measuring   mining   products 
and   in  custom  house  business,  the 
long  ion  of  2240  lb.  is  used. 


Comparative  Table. 
Troy  and  Avoirdupois  Weight. 
1  oz.  Avoir.  =  437^  gr. 
1  lb.      "        =  7000  gr. 
1  oz.  Troy     =  480  gr. 
1  lb.      "        =  5760  gr. 


OF  WEIGHT. 

Troy  Weight. 
24  grains  (gr.)       =  1  penny- 
weight (pwt.). 
20  pennyweights  =  1  ounce. 
12  ounces  =1  pound. 

Troy  weight  is  used  in  the 
measurement  of  the  precious 
metals.  Jewels,  such  as  diamonds 
and  pearls,  are  weighed  by  the 
carat  which  =  3.2  Troy  grains. 
The  term  "carat"  is  also  used  to 
express  the  proportion  of  pure  gold 
in  composition  metals.  If  pure 
gold,  the  metal  is  "24  carats  fine." 
If  one-half  alloy,  it  is  "12  carats 
fine,"  etc. 

Apothecaries'  Weight. 
20  grains      =  1  scruple  (sc.  or  9) . 
3  scruples    =  1  dram  (dr.  or  5)- 
8  drams       =  1  oimce  (oz.  or  5)- 
12  ounces  or 
5760  gr.   =  1  pound  (lb.). 


134  BUSINESS  ARITHMETIC. 

Special  Unit  Weights — Avoirdupois. 
1  bbl.  flour  =  196  lb.  In  most  states: 

1  bbl.  salt  =  280  lb.  1  bu.  potatoes  =  60  lb. 

1  bbl.  pork  =  200  lb.  1  bu.  wheat      =  60  lb. 

1  cu.  ft.  fresh  water   =  62^1b.  1  bu.  com         =  56  lb. 

1  bu.  oats         =  32  lb. 

COUNTING. 

12  things  =  1  dozen  (doz.).  24  sheets  (paper)  =  1  quire  (qr.). 

12  dozen  =  1  gross  (gro.).  20  quires,  or 

12  gross    =  1  great  gross  (g.  gr.).  480  sheets  =  1  ream  (rm.). 

20  things  =  1  score.  (Commercially,   500  sheets  are 

often  used  for  a  ream.) 

MEASURES  OF  EXTENSION. 

Linear  or  Long  Measure.  Surveyor's  Long  Measure. 

12  inches  (in.)  =  1  foot  (ft.).  7.92  inches  =  1  link  (Ik.). 

3  feet                  =  1  yard  (yd.).  25  links        =  1  rod. 

5^  yards,  or  4  rods,  or 

16^  feet          =  1  rod  (rd.).  100  links  =  1  chain  (ch.). 

320  rods,  or  80  chains     =  1  mile. 

5280  feet        =  1  mile  (mi.).  City  property  is  measured  in 

feet  and  decimals  thereof;  large 

Sea  Measure.  tracts  of  farm,  or  unimproved  land 

6  feet              =  1  fathom.  by  surveyor's  measure. 
120  fathoms  =  1  cable  length. 
6086.7  feet,  or  about 

1.15  miles  =  1  knot,  nautical  or 
geographical  mile. 
3  knots  =  1  league. 

Circular  or  Angular  Measure. 
60  seconds  (")  =  1  minute  (').  90  degrees    =  1  right  angle. 

60  minutes        =  1  degree  (°).  360  degrees  =  1  circumference. 

1  minute  =  1  geographic  mile. 

This  measure  is  used  in  geography,  navigation  and  higher  surveying, 
for  the  computation  of  differences  of  time,  locations  on  the  earth's  surface, 
longitude  and  latitude;  and  for  the  measurement  of  angles. 

Square  Measure.  Surveyor's  Square  Measure. 

144  square  inches  =  1  square  foot.  625  square  links  =  1  square  rod. 

(sq.  in.)  (sq.  ft.)  10  square  rod«     =  1  square  chain. 


DENOMINATE   NUMBERS. 


135 


Square  Measure. 


9  square  feet 
30j  square  yards 
160  square  rods 
640  acres 
100  square  feet 


=  1  square  yard. 
=  1  square  rod. 
=  1  acre  (A  ). 
=  1  square  mile. 
=  1  square. 


Surveyor's  Square  Meaeure. 
10  square  chains  =  1  acre. 
640  acres  =  1  square  mile. 

36  square  miles    =  1  township. 


Cubic  Measure. 

1728  cubic  inches  =  1  cubic  foot.  128  cubic  feet  or 

(cu.  in.)  (cu.  ft.)  8  cord  feet       =  1  cord  (wood). 

27  cubic  feet  =  1  cubic  yard.  1  cubic  yard       =  1  load  (earth). 

16  cubic  feet  =  1  cord  foot  (cd.  ft.) . 


MEASURE  OF  TIME. 


60  seconds  (sec.)  =  1  minute  (min.). 

360  days 

=  1  commercial  year. 

60  minutes            =  1  hour  (hr.). 

365  days 

=  1  common  year. 

24  hours                 =  1  day. 

366  days 

=  1  leap  year. 

7  days                   =  1  week  (wk.). 

10  years 

=  1  decade. 

30  days                  =  1  commercial 

100  years 

=  1  century. 

month. 
12  months  (mo.)  =  1  year  (yr.). 

365  days,  5  hours,  48  minutes,  49.7  seconds  =  1  solar  year. 
Centennial  years  divisible  by  400  and  other  years  divisible  by  4  are 
leap  years. 

MEASURES  OF  VALUE. 
U.  S.  Money.  Canadian  Money. 

See  pages  50  and  53.  The  same  as  for  the  United  States. 

100  cents  =  1  dollar  =  $1. 
English  Money. 

4  farthings  (far.)  =  1  penny  (d.). 

12  pence  =  1  shilling  (s.)  =  $0,243  +. 

20  shilUngs  =  1  pound  sterling  (£)  =  $4.8665. 


French  Money. 
100  centimes  =  1  franc 


$0,193. 


German  Money. 
100  Pfennigs  =  1  Mark  =  $0,238. 


Mexican  Money. 
100  centavos  =  1  peso  =  $0,498. 


136  BUSINESS  ARITHMETIC. 

DENOMINATE  NUMBERS. 
INTRODUCTORY    EXERCISE. 

1.  Give  illustration  of  abstract  numbers,  concrete  numbers,  like  and 
unlike  numbers,  denominate  numbers,  simple  and  compound  numbers. 
(See  pp.  1  and  2.) 

2.  Name  simple  denominate  numbers  expressing:  time,  distance,  area, 
value,  weight. 

3.  Name  compound  denominate  numbers  expressing:  cubic  contents, 
liquid  capacity,  area. 

4.  Name  a  very  common  denominate  measure  used  in  each  of  these 
trades,  businesses  or  professions :  grocery,  provision,  oil,  produce,  dry  goods, 
plumbing,  mechanical  engineering,  contracting,  banking,  jewelry,  coal 
and  wood. 

5.  Name  common  denominate  measures  used  in  speaking  of  such 
things  as — 

(1)  The  distance  from  New  York  to  Chicago. 

(2)  The  distance  across  the  street. 

(3)  The  quantity  of  a  meat  order. 

(4)  The  time  required  to  do  an  errand. 

(5)  The  amount  of  imports  of  the  United  States  for  last  year. 

6.  Name  the  common  measures  used  in  your  household  and  its  trading. 

7.  What  is  the  advantage  of  buying  by  some  measure  or  fraction  of  it, 
over  buying  by  two  or  more  denominations  of  a  thing? 

8.  What  are  the  advantages  to  the  retailer  in  selling  goods  in  de- 
nominate packages,  rather  than  in  bulk? 

It  is  evident  from  the  tables  that  a  number  expressed  in  one 
denomination  may  be  expressed  in  another  denomination  of 
the  same  table,  and  sometimes  of  other  tables,  without  altering 
its  value.  The  process  of  changing  the  form  of  expression  of 
a  quantity  is  termed  reduction  ascending,  if  the  denomination 
chosen  is  larger  than  the  original,  and  reduction  descending,  if  it 
is  smaller. 

REDUCTION  DESCENDING. 

Illustrations.     (1)  Reduce  10  rds.  3  yd.  to  feet. 
Analysis  and  Solution,     (a)  1  rd.  =  5.5  yd.     10  rd.  =  55  yd.     55  yd. 
+  3  yd.  =  58  yd.     1  yd.  =  3  ft.    58  yd.  =  3  X  58  ft.  or  174  ft. 


DENOMINATE   NUMBERS. 


137 


(6)   1  rd.  =  16.5  ft.      10  rd.  =  10  X  16.5  ft.  or  165  ft. 
3  yd.  =  3  X  3  or  9  ft.     165  ft.  +  9  f t.  =  174  ft. 
(2)  Reduce  .45  gal.  to  lower  denominations. 
Analysis  and  Solution. 

1  gal.  =  4  qt.  .45  of  4  qt. 

1  qt.    =2  pt. 
1  pt.    =4  ^. 

.-.  .45  gal.  =  Iqt.  1  pt.  2.4  gi. 


1.8  qt. 
.8  of  2  pt.  =  1.6  pt. 
.6    of  4  gi.   =  2.4  gi. 


1  yd  =  3  ft. 


Find  missing  values: 

1.  3  gal.  = qt. 

2.  2  cd.  = cu.  ft. 

3.  2  sh.  6  d.  = d. 

4.  2\  gal.  = pt. 

5.  3/4  bu.  = pt. 

6.  2  m.  = rd. 

7.  2  bu.  = pt. 

8.  1  gr.  3  doz.  = units. 


ORAL    EXERCISE. 


9. 

.4  pk.  = qt. 

10. 

3.5  sq.  yd.  = sq.  f 

11. 

U'  6"  = ". 

12. 

65  score  = units. 

13. 

1  mi.  40  rd.  = rd. 

14. 

1.3  T.  = lb. 

15. 

£2.25  = sh. 

EXERCISE. 


3.  5.6  gal.  to  gills. 

4.  1.375  sq.  mi.  to  A. 


Reduce: 

1.  7.3  cu.  yd.  to  cu.  in. 

2.  31  T.  to  oz. 
6.  7.475  mi.  to  ft. 

6.  Find  the  cost  of  5f  A.  @  5c  per  sq.  ft. 

7.  What  will  it  cost  to  excavate  942.5  cu.  yd.  @  15c  per  cu.  ft.? 

8.  Express  2  mi.  in  equivalent  engineer's  unit  of  1000  ft. 

9.  Reduce  11  gt.  gross  to  a  fraction  of  1000  units. 

10.  What  part  of  1  rd.  are  6  in.?     11.    What  part  of  1  mi.  are  2  y4.? 


REDUCTION  ASCENDING. 

Illustrations.     (1)  Express  729  ft.  in  higher  denominations. 
Solution.    Since  3'  =  1  yd.  3)729 

1  rd.  =  5i  ft.  5.5)243  =  no.  yd. 

(mult,  by  .2)  or  11)486 

729  ft.  =  44  rd.  2  yd.  44  =  no.  rd. 

2  yd.  rem. 


138  BUSINESS  ARITHMETIC. 

(2)  Express  2  ft.  9  in.  as  a  decimal  of  a  yard. 
Solution.     ■:  2  ft.  9  in.  =  33  in.  and  1  yd.  =  36  in. 

.-.  2  ft.  9  in.  =  33/36  of  a  yd.    33  -^  36  =  .916  +. 

.-.  2  ft.  9  in.  =  .916+  yd. 

ORAL    EXERCISE. 

Reduce  to  the  next  higher  denomination: 


1.     52  ft.  '                     3. 

12  yd. 

5.     65  in.           7.     8^  sh. 

2.     27i  pt.  (dry).           4. 

27,000  lb. 

6.     148  oz.         8.     50  cu.  ft. 

Find  the  cost  of: 

9.    2  lb.  @  6c  per  oz. 

13. 

3i  gal.  @  5c  per  pt. 

10.     5  qt.  @  8c  per  pt. 

14. 

3  pk.  @  60c  per  bu. 

11.     4  yd.  @  5c  per  in. 

15. 

40  ft.  @  30c  per  yd. 

12.    2  bu.  @  30c  per  pk. 

16. 

900  lb.  @  $36  per  T. 

EXERCISE. 

(Decimals  to  4  places.) 

1.  Express  as  decimals  of  a  mile:  1  rd.j  1  ft.;  1  yd.;  7.35  in.;  8.4  yd. 

2.  Express  as  decimals  of  a  bu.:  1  pk.;  3  qts.;  1  pt.;  3^  pt. 

3.  Express  as  decimals  of  a  cu.  yd.:  1  cu.  ft.;  20  cu.  in. 

4.  Express  as  decimals  of  a  £:  1  sh.;  1  d.;  7  sh.  6  d.;  £  3  8  sh.  5  d. 

5.  Express  as  decimals  of  a  T. :  11  lb.;  1  oz.;  745  lb.;  212  lb.  8  oz. 

6.  Express  as  decimals  of  an  acre :  1  sq.  rd. ;  sq.  yd. ;  96  sq.  ft. ;  45  sq.  yd. 

7.  Reduce  to  higher  denominations:  326,425  in.;  8429  ft.  (dry); 
11,927  cu.  in 

8.  What  quantity  (gallons)  of  ohve  oil  must  be  bought  to  fill  90  gross 
of  pint  bottles? 

The  Fundamental  Processes.  The  fundamental  pro- 
cesses, as  applied  to  denominate  or  compound  numbers,  differ 
relatively  little  from  the  same  processes  in  simple  numbers. 
The  main  point  of  difference  is  in  the  substitution  of  an  irreg- 
ular reduction  scale  for  the  decimal  scale. 

ADDITION. 

Illustration.  (1)  Add  567  and  329.  (2)  Add  5  bu.  3  pk.  5  qt.  and 
3  bu.  2  pk  4  qt. 


DENOMINATE   NUMBERS.  139 

Sohdions.      (1)  (2)  (2)  Condensed. 

567        5  bu.  3  pk.  5  qt.  5  bu.  3  pk.  5  qt. 

329        3         2         4  3         2         4 

896        8  bu.  5  pk.  9  qt.  9  bu.  2  pk.  1  qt. 

or  9  bu.  2  pk.  1  qt.       (Reduction  ascending.) 
Note.     (2)  In  the  condensed  form  the  reduction  is  performed  mentally, 
as  the  total  of  each  measure  is  obtained.     Thus  the  total  of  the  quart 
column  is  9  qt.     Since  9  qt.  =  1  pk.  1  qt.,  the  1  quart  is  written  in  the  sum, 
and  the  1  peck  is  carried  forward  to  the  next  column. 

ORAL    EXERCISE. 


Find  the  sum 

of: 

1. 

2. 

3. 

5  gal.  2  qt.  1  pt. 

6  bu.  1  pk. 

1  T.  12  cwt. 

2         1        1 

13- 

3 

5qt. 

3         9 

4. 

5. 

6. 

15  cwt.  84  lb. 

7] 

mi.    46  rd. 

6sq. 

yd.  5  sq.  ft. 

7          19 

2^ 

300 

1 
2 

1 
8 

7.    To  3  gal.  add  consecutive  pints.  % 

EXERCISE. 

1.  Find  the  total  if  the  items  of  a  purchase  from  a  London  merchant  are : 
£5  3  s.  6  d.;  £  18  11  s.  4  d.;  £12  5  s.  8  d. 

2.  In  St.  Clair  County  the  following  stretches  of  country  road  have 
been  macadamized  in  the  past  year.  1  mi.  160  rd.,  2  mi.  145  rd.,  8  nai. 
176  rd.,  1  mi.  46|  rd.,  a  total  of  ?. 

3.  A  lumberman,  who  already  owns  stumpage  rights  on  3  sq.  mi. 
260  A.,  buys  rights  to  neighboring  tracts  of  84  A.,  1  sq.  mi.  52  A.,  526  A. 
40  sq.  rd.,  and  64  A.  120  sq.  rd.  He  then  has  rights  to  ?  sq.  mi.  ?  A.  or 
to  ?  A.  For  the  rights  to  lumber  these  new  tracts,  he  paid  $85  per  A.  or 
$  ?  m  aU. 

SUBTRACTION. 
Subtraction  involves  reduction  descending. 

Illustrations.  (1)  (2) 

5  bu.  3  pk.  6  qt.  5  T.  7  cwt.  16  lb. 

2         1         3  2       5  46 

Differences  3  bu.  2  pk.  3  qt.  3  T.  1  cwt.  70  lb. 

Reduce    1    cwt.  of    the    minuend  to    pounds.     116   lb.  —  46    lb.  = 
70  lb.;  6  cwt.  —  5  cwt.  =  1  cwt.,  etc. 


140                        BUSINESS  ARITHMETIC. 

ORAL  EXERCISE. 
Find  the  difference  between : 

1.                        2.  3.                        4. 

5  bu.  3  pk.        5  gal.  2  qt.  16  cwt.18  lb.        5  lb.  11  oz. 

12               23  4         20              32 


Find  the  difference  between  the  larger  and  smaller  quantity  in  ex.  1-5 
of  the  last  oral  exercise. 

EXERCISE. 

1.  What  check  can  you  suggest  for  compound  addition  and  subtraction? 
Illustrate  with  examples. 

2.  On  a  contract  for  the  laying  of  752  cu.  yd.  of  masonry,  15  cu.  yd. 

7  cu.  ft.  are  laid  one  day,  31  cu.  yd.  14  cu.  ft.  the  second  day,  68  cu.  yd. 

8  cu.  ft.  the  third  and  88  cu.  yd.  11  cu.  ft.  the  fourth.     How  much  work 
remains  to  be  done? 

3.  From  15.0739  mi.  subtract  2  mi,  126  rd.  2  yd. 

4.  Make  the  extensions  in  this  form: 

Gross  Cost.  Discount.  Net  cost. 

£  16—5—  8  £  2—  6—4            £ 

4—9—  3  8—7                

84—7—11  3—15—8  ————■ 

Totals             ?  ?  ? 

MULTIPLICATION. 
That  no  new  principle  is  involved  is  seen  from  this 
Illustration.    Find  the  product  of  3  bu.  2  pk.  1  qt.  by  8. 
Solution.  Analysis. 

3  bu.  2  pk.  1  qt.  8  X  1  qt,  =  8  qt.  =  1  pk. 

S_  8  X  2  pk.  +  1  pk.  =  17  pk.  =  4  bu.  1  pk. 

28  bu.  1  pk.  0  qt.  8  X  3  bu.  +  4  bu.  =  28  bu. 

EXERCISE. 

(Solve  mentally  if  possible.) 
Multiply: 

1.  3  ft.  2  in.  by  6. 

2.  1  yd.  2  ft.  by  8. 

3.  3  lb.  2  oz.  by  30. 

4.  2  pk.  3  qt.  by  12. 
9.    How  much  silver  is  required  for  five  ornaments  each  containing 

1  oz.  7  pwt.? 


5. 

2  qt.  1  pt.  by  8. 

6. 

3  rd.  27  Unks  by  9. 

7. 

3  cwt.  60  lb.  by  10. 

8. 

1  T.  4  cwt.  by  20. 

DENOMINATE   NUMBERS.  141 

10.  A  factory  manager  reduces  the  length  of  a  metal  bar  in  a  machine 
he  manufactures  3f  in.  This  means  a  saving  of  how  much  metal  in 
12,000  bars? 

11.  Determine  the  cost  of  75  yd.  extra  broadcloth,  purchased  at  £1 
2  8.  3  d.  per  yd. 

12.  A  factory  expert  discovers  that  the  length  of  a  bolt  used  in  the 
company's  product  may  be  reduced  1/4  in.  This  means  a  saving  of  how 
many  feet  if  250,000  bolts  are  used  per  year? 

13.  Compute  the  cost  of  a  car  load  containing  60  bbl.  of  apples,  each 
containing  2  bu.  2  pk.,  at  75c  per  pk. 

DIVISION. 

In  division  of  denominate  numbers  the  divisor  may  be  either 
abstract  or  denominate. 

Illustrations.  (1)  Compute  the  cost,  per  yd.,  of  80  yd.  cloth  invoiced 
at  £180. 

Solution  (a).  Solution  and  Analysis  (5). 

80)  £180  The  cost  per  yd.  is  evidently  1/80  of 

£    2.25  =  £2  5  8.  (cost)  £180.         1/80  of    £180  =  £2    and   a 

remamder     of     £20.     £20  =  400     s. 
1/80    of    400    8.  =  5    8.     .*.    1/80    of 
£180  =  £2  5  8. 
(2)  At  a  price  of  £1  2  s.  per  yd.,  how  many  yards  may  be  purchased  for 
£89  2  8.? 

Solution. 
£  1  2  s.  =      22  8.  The  numbers  being   concrete,   re- 

£89  2  s.  =  1782  s.  duction  to  a  common  denominator  is 

81    =  no.  of  yd.  necessary. 

~22)1782 

ORAL   EXERCISE. 
Divide : 

1.  6  bu.  3  pk.  by  3. 

2.  5  gal.  2  qt.  by  2. 

3.  3  T.  6  cwt.  by  200. 

4.  41b.  by  8. 

5.  3  pk.  6  qt.  by  10.  11.     5  bu.  1  pk.  by  1  bu.  3 

6.  5  yd.  1ft.  by  4. 


7. 

3  sq.  yd.  3  sq.  ft.  by  6. 

8. 

3  gal.  by  2  qt. 

9. 

4  lb.  by  8  oz. 

10. 

2  ft.  6  in.  by  3  m. 

142  BUSINESS   ARITHMETIC. 

EXERCISE. 

1.  Divide  960  cu.  yd.  5.4  cu.  ft.  by  11.5,  checking  by  multiplication. 

2.  A  carriage  wheel  revolves  two  times  in  7  yards,  or times  in  one 

mile. 

3.  Divide  5  bu.  7  pk.  3  qt.  by  1  pk.  2  qt.  1  pt.  Check  the  result  by 
multiplication. 

4.  A  dealer  has  2  bbl.,  31|  gal.  each,  of  olive  oil,  to  be  bottled  in 
half-pint  bottles.     He  requires  —  bottles. 

5.  A  newspaper  press  printed  57,384  copies  in  2  hr.  5  min.,  at  the 
average  rate  of  —  copies  per  minute. 

Individual  Original  Work. 

Report  on  "Common  Measures  Used  in  Business."  Disregard  text- 
book classifications,  and  find  out  what  parts  of  tables  are  in  actual  common 
use.  Classify  under:  (1)  Common  to  business  in  general;  (2)  limited  to 
particular  trades;  (3)  non-text-book  measures.     Give  illustrative  examples. 


CHAPTER   XXIII. 

PRACTICAL  MEASUREMENTS. 

Many  simple  measurements  in  business  and  science  are 
based  on  elementary  geometric  principles.  A  slight  knowledge 
of  some  of  these  principles,  and  of  the  geometric  figures  to 
which  they  apply,  is  of  value  to  everyone. 

Geometric  Conceptions — Plane  Figures. 

A  geometric  line  is  considered  to  have  extension  but  neither 
length  nor  breadth.  A  straight  line  is  the  shortest  distance 
between  two  points.  Lines  are  parallel  if  they  are  the  same 
distance  apart  throughout  their  entire  length.ZZ 

An  angle  is  the  divergence  of  two  lines  having  a  common 
point.     AXB  is  the  angle  of  the  lines  AX  and  XB,     If  the 


two  lines  meet  so  that  the  two  angles  on  the  same  side  of  one 
line  are  equal,  the  angles  are  right  angles,  and  the  lines  are 
perpendicular.  Thus  AYC  and  BYC  are  right  angles  and  CY 
is  perpendicular  to  AB.  An  acute  angle  is  less  than  a  right 
angle.     An  obtuse  angle  is  greater  than  a  right  angle. 

A  geometrical  surface  has  length  and  breadth  but  no  thick- 
ness. A  plane,  or  plane  surface,  is  a  level  surface,  such  as  that 
of  still  water.     A  figure  in  a  plane  is  a  plane  figure. 

143 


144 


BUSINESS   ARITHMETIC. 


A  triangle  is  a  plane  figure  bounded  by  three  straight  lines. 
It  is  called  equilateral,  isosceles,  or  scalene,  according  as  it  has 


three  sides  equal  (A),  two  sides  equal  {B),ot  no  sides  equal  (C). 
A  right  angled  triangle  has  one  right  angle  (D). 


A  quadrilateral  is  a  plane  figure  bounded  by  four  straight 
lines.  If  the  opposite  sides  of  a  quadrilateral  are  parallel,  the 
figure  is  called  a  parallelogram  (E-H).  The  parallelogram  is 
a  rectangle  if  it  has  four  right  angles  (F).     It  is  a  square  if  the 


sides  are  equal  and  the  angles  right  angles  (G).  It  is  a 
rhombus,  if  the  four  sides  are  equal  but  the  angles  are  not 
right  angles  (H), 


PRAlCTICAL   MEASUREMENTS. 


145 


A  circle  is  a  plane  figure  bounded  by  a  curved  line,  called  the 
circumference,  every  point  of  which  is  equidistant  from  a  point 
within  called  the  center.  The  diameter 
is  any  straight  line  passing  through 
the  center  and  terminating  in  the  cir- 
cumference. The  radius  is  one-half 
of  the  diameter.  An  arc  is  any  part 
of  the  circumference,  and  is  measured 
by  degrees  of  angular  measure. 

The  circumference  is  approximately 
3.1416  (approximately  3y)  times  the 
diameter.      This    factor  is    called    by 

the  Greek  letter  ''tt,"  (pronounced,  "pi.")  If  the  diameter 
is  known,  the  circumference  may  be  determined  by  multiply- 
ing it  by  3.1416.  If  the  circumferenpe  is  known,  the  diameter 
is  determined  by  finding  the  quotient  of  the  circumference 
divided  by  3.1416. 

The  perimeter  of  a  plane  figure  is  the  distance  around  it. 
The  ha^e  is  the  side  on  which  it  is  assumed  to  rest.  The 
altitude  is  the  perpendicular  distance  from  the  base  to  the 
most  distant  point  of  the  figure. 


EXERCISE. 

1.  Draw  four  circles  of  4",  5",  6",  and  8"  radius,  respectively.  In 
each  case,  measure  carefully  the  circumference  and  divide  by  the  re- 
spective diameters.  .  Take  the  average  of  the  resulting  quotients.  How 
close  to  the  value  of  "x"  do  you  approach?  By  computation,  find  your 
error  in  measurement  of  each  circumference.  To  what  is  this  error  due? 
11 


146 


BUSINESS  ARITHMETIC. 


2.  Compare  the  diameters  and  circumferences  of  two  circles,  one  of 
which  has  a  diameter  of  10'  and  the  other  a  circumference  of  42'. 

3.  What  is  the  diameter  of  a  circular  path  a  mile  in  circumference? 

4.  Compute  the  cost  of  the  fencing  around  a  circular  park  of  90  ft. 
radius,  at  $1.50  per  foot. 

5.  In  racing  on  a  circular  track,  what  gain  may  result  from  securing 
an  inside  position?     Illustrate. 

6.  A  man  has  1000  ft.  of  movable  fencing  for  a  chicken  nm.  Give 
dimensions  of  a  square  field,  a  rectangular  field  and  a  circular  field  that 
he  may  enclose. 

Areas  of  Plane  Figures. 

The  accompanying  rectangle  is  divided  by  parallel  lines  into 

unit  squares.     How  many  unit  squares  in  each  row?     How 

many   rows?     Which    dimension    shows 

the  number  of  rows?     Which  the  units 

per  row?     What  is  the  total  number  of 

units?      How  obtained?      Show  that  the 

number  of  square  units  would  be   the 

same  if  "5''  were  considered   the   base 

and  "6"  the  altitude.     It  is  evident  that 

the  area  of  the  rectangle  is  equal  to  the 

product  of  its  length  and  width — its  two  dimensions. 

In  the  case  of  the  parallelo- 
gram,by  cutting  off  the  section 
X  and  placing  it  in  the  posi- 
tion X',  it  is  evident  that  the 
figure  is  equivalent  in  area  to 
a  rectangle  having  the  same 

base  and  an  equal  altitude.     Try  this  experimentally  with  a 
piece  of  paper. 

In  the  case  of  the  triangle,  as  shown  by 
the  figure,  the  area  is  equal  to  one-half 
the  area  of  the  rectangle  erected  on  the 
same  base,  and  with  equal  altitude.  The 
area  is  equal,  therefore,  to  one-half  the  pro- 
duct of  its  base  and  altitude. 


PRACTICAL  MEASUREMENTS.  147 

If  a  circle  is  divided  as  shown,  and  the  parts  rearranged  as  a 
series  of  approximate  triangles,  it  is  evident  that  the  area  will 
equal  the  sum  of  the  areas  of  the  triangles.  If  the  triangles 
are  made  small  enough  they  have  altitudes  equal  to  the  radius 
of  the  circle.     The  sum  of  their  bases  equals  the  circumference 


of  the  circle.  The  area  of  the  circle,  therefore,  equals  one- 
half  the  product  of  the  radius  by  the  circumference.  Since 
the  circumference  equals  tt  X  the  diameter  (2  X  radius)  the 
area  equals  ^RX  2RX  3.1416,  or  3.1416  X  the  square  of  the 
radiiLS. 

The  area  of  irregular  right  line  figures  may  be  obtained  by 
dividing  them  into  triangles  and  parallelograms  and  deter- 
mine the  areas  of  these  parts. 

ORAL  EXERCISE. 
Compute  the  areas  of  the  following: 

Figure.  Altitude.  Base. 

1.  Triangle  14  ft.  16  ft. 

2.  Rectangle  24  ft.  2^  ft. 

3.  Square  15  ft.  

4.  Parallelogram  16  in.  2  ft. 

5.  Rectangle  2\ii.  4Ht. 

6.  Triangle  20  in.  1  ft.  6  in. 

7.  Parallelogram  6i  ft.  2  ft.  3  in. 

8.  Find  the  difference  between  3  sq.  yd.  and  an  area  3  yd.  square. 

9.  How  does  a  5  in.  square  compare  with  an  area  of  5  sq.  in.? 

10.     Name  two  factors  of  360.     Find  two  dimensions  for  a  rectangle  of 
360  sq.  ft.  area.     Find  other  dimensions  for  a  second  rectangle  of  the  same 


148 


BUSINESS   ARITHMETIC. 


area.     How  many  rectangles  have  the  same  area?     If  one  dimension  is 
4'  what  is  the  other?     If  one  dimension  is  12  ft.  what  is  the  other? 

11.  A  series  of  rectangles  have  a  common  area  of  480  sq.  ft.  Find  the 
bases  if  the  altitudes  are  respectively  4,  16,  12,  6,  8,  15,  20,  40,  1/2  feet. 
Compare  the  perimeters.  Do  rectangles  of  the  same  area  have  the  same 
perimeter? 

12.  Give  the  dimensions  of  several  rectangles  that  have  a  common 
area  of  1200  sq.  in. 

13.  To  what  is  the  area  of  a  triangle  equal?  If  the  area  and  one 
dimension  are  known,  how  is  the  other  determined? 

14.  Find  the  altitudes  of  a  series  of  triangles  having  a  common  area 
of  180  sq.  ft.  and  bases  of  4,  3,  20,  9,  18  and  30  ft.,  respectively. 

15.  Name  the  bases  and  altitudes  of  ten  triangles  that  have  a  common 
area  of  720  sq.  in. 

EXERCISE. 


1.  Determine  the  area  of  a  floor  measuring  25  ft.  4  in.  by  16  ft.  8  in. 

2.  Determine  the  area  of  the  end  section  of  a  metal  bar  measuring 
3j  in.  by  1|  in. 

3.  How  many  3-in.  squares  may  be  cut  from  a  piece  of  card  board 
measuring  25  in.  by  36  in.? 

4.  What  is  the  acreage  of  a  field  measuring  720  rd.  by  96  rd.? 

5.  A  field  measuring  88  rd.  by 
126  rd.  produces  280  bu.  of  grain, 
an  average  of bu.  per  acre. 

6.  Subdivide  the  field  repre- 
sented by  the  accompanying  dia- 
gram into  rectangles,  as  shown  by 
the  dotted  lines.  Name  the  dimen- 
sions of  each  rectangle.  Determine 
the   total   area.     Check  the    com- 

10/  putation    by  subdividing  into  an- 

other   group    of    geometric  figures 
and  determining  area. 

7.  How  many  oblongs,  2  in.  by  3^  in.,  can  be  cut  from  sheets  of 
composition  metal  24  in.  by  36  in.?  Does  it  make  any  difference  which 
way  the  oblongs  are  laid  off  on  the  sheet? 


12' 

12' 

6'- 

to 

•o 

PRACTICAL   MEASUREMENTS. 


149 


80' 


8.  The  diagram  here  shown  rep- 
resents a  park  with  a  circular  flower 
bed.  The  walk  is  5  ft.  wide.  Find 
the  area  of  the  flower  bed.  Find 
the  number  of  square  yards  of  sod- 
ding required  for  the  parking. 
How  is  this  area  determined  from 
the  area  of  the  rectangle  and  the 
area  of  the  circle? 

Compute  the  cost  of  constructing 
the  walk,  at  $1.25  per  sq.  yd.   Com- 
pute the  cost  of  curbing  for  the  inner  and  outer  edges  of  the  walk,  at  38c 
per  linear  foot. 

9.  Find  the  acreage  of  the  plot  of  land  here  represented .   Express  dimen- 

sions in  linear  measure.  Check 
computation  of  acreage  by  using 
dimensions  just  found. 


Applications  of  Square 
Root. 

Square  root  frequently  is 
employed,  in  connection 
with  geometric  figures,  in  de- 
termining missing  dimen- 
sions. 


1.  Squares.  Since  the  area  of  a  square  is  equal  to  the 
product  of  its  two  equal  sides,  it  is  evident  that  the  length 
of  one  side  is  equal  to  the  square  root 

of  the  area. 

2.  Right  Triangles.  It  has  been  proved, 
geometrically,  that  the  square  of  the 
hypotenuse,  or  side  opposite  the  right 
angle,  is  equal  to  the  sum  of  the  squares 
of  the  other  two  sides.  This  is  illus- 
trated graphically  in  the  drawing.  It 
is  evident,  then,  that  the  length  of  the 
hypotenuse  is  equal  to  the  square  root 


150 


BUSINESS   ARITHMETIC. 


of  the  sum  of  the  squares  of  the  other  two  sides;  and  that  the 
length  of  either  other  side  is  equal  to  the  square  root  of  the 
difference  of  the  squares  of  the  hypotenuse  and  the  known  side. 
3.  Since  the  area  of  a  circle  equals  tt  X  the  square  of  the 
radius,  it  is  evident  that  the  radius  equals  the  square  root 
of  the  area  divided  by  3.1416. 


EXERCISE. 

1.  Determine  the  side  of  a  square  having  the  same  area  as  a  rectangle 
measuring  54  rd.  by  17.3  rd. 

2.  To  have  an  area  of  60,000  sq.  ft.,  a  square  must  have  an  edge  of 
how  many  feet? 

3.  Find  the  edge  of  a  square  that  has  the  same  area  as  a  circle  of 
50  ft.  radius. 

4.  Determine  the  radius  of  a  circle  that  has  the  same  area  as  a 
square  of  80  ft.  side. 

5.  How  many  linear  feet  of  fencing  are  required  for  a  rectangular 
chicken  run,  80  ft.  by  120  ft.?  What  is  the  difference  in  amount  of  fencing 
required  for  a  square  run  of  the  same  area? 

6.  Compare  the  perimeters  of  a  square,  a  rectangle  of  20  ft.  altitude, 
and  a  circle,  if  all  have  equal  areas  of  600  sq.  ft. 

7.  If  the  two  sides  of  a  right  triangle  are  8  ft.  and  16  ft.,  what  is  the 
length  of  the  hypotenuse? 

Determine  the  missing  values  in  the  figures  shown  below: 

8.  10. 


PRACTICAL  MEASUREMENTS. 


151 


Gables. 

The  pitch  of  a  roof  is  the  amount  of  rise  of  a  rafter  for  each 
foot  in  the  base  of  the  gable. 

Illustkation.     In  the  cut,  the  rafter  rises  10  ft.  for  a  base  of  20  ft. 
This  is  a  pitch  of  10/20,  or  1/2.     The  familiar  Gothic  pitch  is  5/8. 


EXERCISE. 

Fmd  the  missing  values: 

Base  Width.    Height. 
1.    30  ft.            18  ft. 

Pitch. 
? 

Length 
? 

of  Rafter. 

(Allow  1  ft.  Overhang.) 

2.     40  ft. 

? 

Gothic 

? 

(<                H                       11 

3.    45  ft. 

20  ft. 

? 

? 

"    li  ft. 

4.        ? 

16  ft. 

1/4 

? 

(t         tl            tt 

5.    36  ft. 

? 

1/2 

? 

tt               U                     tl 

PRACTICAL  APPLICATIONS  OF  SQUARE  MEASURE. 
The  following  applications  are  selected  to  illustrate  the  use 
of  square  measure  as  modified  by  business  custom. 

1.    Flooking. 

Considerable  structural  work,  such  as  flooring,  roofing, 
tiling,  etc.,  is  measured  frequently  by  the  square,  or  100  sq.  ft. 
More  rarely  the  measurement  is  expressed  directly  in  sq.  ft., 
or  in  units  of  1000  sq.  ft. 

In  properly  matching  and  fitting  tongue  and  groove  flooring, 
there  is  considerable  waste  of  material.     Thus,  for  every  foot 


152 


BUSINESS  ARITHMETIC. 


of  flooring  in  place,  1^  to  1^  ft.  of  flooring  material  may  be 
required. 

EXERCISE. 

1.  Compute  the  cost  of  flooring  a  hall  measuring  80'  by  35',  at  $2.95 
per  square. 

2.  Compute  the  cost  of  flooring  with  hard  wood,  the  two  front  rooms 
shown  in  this  floor  plan  of  a  bungalow,  at  $40  per  M,  allowing  1/3  waste, 
and  $10.50  for  extras. 


DINING  ROOM 


/   ^    .o'«'/ 


10'  X  12'6 


BED  ROOM 


JO'  X  10  6' 


-*     -r-i b^ 


3.  How  many  tile,  6"  square,  are  required  for  the  right  rear  room, 
allowing  1/20  extra  for  breakage? 

4.  A  gable  roof,  42'  long,  and  having  rafters  21'  long,  is  to  be  covered 
with  roofing  tile.  At  the  rate  of  300  per  square,  how  many  tile  are  re- 
quired? How  many  shingles  are  required  for  the  same  purpose,  at  the 
rate  of  800  per  square?     How  many  bundles  of  shingles,  250  each? 

Note.  Carpenters  and  builders  have  tables  showing  the  numbers 
of  tile,  shingles,  etc.,  of  different  sizes,  req^uired  per  square.  These  are 
often  used  as  a  basis  of  computation  in  practical  work. 


PRACTICAL   MEASUREMENTS.  153 

2.     Plastering. 

Plastering  is  commonly  measured  by  the  square  yard. 
Allowances  for  openings  are  made  in  various  ways,  such  as 
(1)  by  actual  measurement,  (2)  by  arbitrary  allowance,  etc. 
If  working  plans  for  a  building  are  not  given,  dimensions 
of  rooms  are  usually  stated  in  this  order:  Length — width — 
height. 

Example:  21'  X  16'  X  9'  6''. 

EXERCISE. 

1.  Compute  the  cost  of  hard  finishing  the  walls  and  ceiUng  of  the  two 
front  rooms  of  the  building,  page  152,  at  43c  per  sq.  yd.  Height  of  room 
9'.     Allow  20  sq.  ft.  for  door  openings  and  18  sq.  ft.  for  windows. 

2.  Compute  the  cost  of  rough  finishing  the  bedroom  and  dining  room 
at  26c.  per  sq.  yd.,  allowing  by  actual  measurement  for  the  doors,  which 
are  1'  by  4',  and  for  windows,  which  are  6'  by  4^'. 

3.    Painting. 

Painting  is  usually  measured  by  the  square  yard,  no  allow- 
ance being  made  for  windows  or  openings  of  a  similar  nature. 

EXERCISE. 

1.  Compute  the  cost  of  painting  a  fence,  6'  high,  around  the  lot  shown 
on  page   148,  both  sides,  at  a  cost  of  24c  per  sq.  yd. 

2.  Compute  the  cost  of  painting  with  two 

coats  the  walls  of  a  building  48'  long,  and  having  ^-^''^'^N^ 

this  end  section.     Price  for  double  coat,  45c  per 
sq.  yd. 

4.    Papering. 

Although  dimensions  vary,  the  width 
of  wall  paper  is  usually  assumed  to  be  18 
inches.     The  length  is  8  yards  for  single 
rolls  and  16  yards  for  double  rolls.     Complete  rolls  must  be 
bought. 

Allowance  is  made  for  openings  in  various  ways,  such  as 


CEILING    PLAN 


154  BUSINESS   ARITHMETIC. 

by  (1)  exact  measurement;  (2)  uniform  allowance  for  each 
opening;  (3)  allowance  for  width  of  opening,  and  full  height  of 
room,  the  assumption  being  that  remnants  of  rolls,  after 
cutting  full  strips,  will  be  sufficient  to  paper  the  wall  over 
and  under  openings;  and  (4)  an  arbitrary  allowance  according 
to  area,  as  "  one  roll  to  24  sq.  ft.  regardless  of  openings.'* 

ORAL    EXERCISE. 

1.     If  paper  is  laid  on  the  ceiling  represented  in  the  accompanying  il- 
lustration parallel  to  the  shorter  dimension,  what  is  the  length  of  each  strip 
in  yards?     How  many  strips  can  be  cut  from  each 
24  FT.  double  roll?     How  many  strips  are  needed?  How 

many  rolls?     What  is  the  cost,  at  45c  per  roll? 

2.  The  walls  of  the  room,  the  ceiling  of  which 
is  shown,  are  also  to  be  papered.  What  does  the 
border  cost,  at  15c  per  yd?  The  strips  of  papering 
on  the  walls  must  be  8  ft.  long.  How  many  strips 
are  required?  How  many  strips  can  be  cut  from 
each  double  roll?  How  many  rolls  are  needed? 
What  is  the  cost,  at  50c  per  roll? 

It  is  evident  that  the  simplest  method 
of  approximation  is  to  divide  the  perime- 
ter of  the  room,  less  the  width  of  all  openings,  by  the  width  of 
a  strip,  and  to  divide  the  number  of  strips  by  the  number 
of  complete  strips  in  one  roll,  in  order  to  determine  the 
number  of  rolls. 

EXERCISE. 

1.  A  room  30'  by  28'  has  five  openings,  each  2\'  wide,  and  one 
opening  4'  wide.  It  is  9'  from  baseboard  to  ceiling.  Find  the  cost  of 
papering  sides  and  ceiling,  without  border,  at  55c  per  roll,  allowing  6" 
per  strip  for  matching  design. 

2.  Compute  the  quantity  of  ceiling,  border  and  wall  paper  required 
for  your  class  room. 

3.  Find  the  quantity  of  wall,  ceiling  and  border  paper  required  for 
each  of  the  rooms  shown  in  the  diagram  on  page  152,  assuming  the  wall 
height,  allowing  for  border,  as  9',  and  allowing  6"  per  strip  for  matching 
design.     Assume  width  of  doors  3';  of  windows,  2\'. 


PRACTICAL   MEASUREMENTS.  155 

5.     Carpeting. 

Carpeting  is  measured  by  the  yard  'per  length  of  strip, 
regardless  of  width.  Oil  cloth,  linoleum  and  carpet  linings 
are  sometimes  measured  by  the  square  yard. 

In  order  to  compute  the  quantity  of  carpet  for  a  room,  it  is 
necessary  to  determine  the  number  of  strips  and  their  length, 
somewhat  after  the  manner  of  the  computations  for  papering. 
The  number  of  strips  depends  on 
the  direction  in  which  the  strips 
are   laid.      Allowance    must   be 
made,  also,  for  matching  design. 

Illustration.  Compute  the  quan- 
tity of  carpet  required  for  the  room 
shown,  if  laid  lengthwise,  the  width  of 
the  carpet  being  three  feet. 

Solution.  19-T-3  =  6^,  no.  of  strips. 
Therefore,  7  strips  are  required.  24'  = 
length  of  one  strip  =  8  yd.  7X8  =72, 
no.  of  yd.  required. 

Note.  Since  each  strip  is  3'  wide,  the  total  width  of  19'  divided  by  3 
will  give  the  number  of  strips,  a  full  strip  being  purchased  for  each  fraction 
of  width. 

EXERCISE. 

1.  Find  the  quantity  of  carpet  required  for  the  room  shown  above  if 
laid  across  the  shorter  dimension? 

2.  Compute  the  cost  of  Brussels  carpet,  27"  wide,  for  the  above  room, 
laid  as  shown,  at  $1.50  per  sq.  yd.,  allowing  6"  per  strip,  after  the  first 
strip,  for  matching.  Why  need  no  allowance  be  made  for  matching  the 
first  strip? 

3.  The  above  robm  is  to  be  carpeted  with  Ingrain  carpet,  1  yd.  wide, 
laid  lengthwise,  and  with  a  border  around  the  room  2'  wide.  Determine 
the  quantity  of  border  and  carpet,  allowing  1/20  for  matching.  Compute 
the  number  of  square  yards  of  lining  required. 

4.  Find  the  cost  of  carpeting  the  dining  room  (page  152)  with  linoleum, 
1^  yd.  wide,  at  $1.45  per  sq.  yd.,  allowing  2  sq.  yd.  for  matching  design. 

5.  Compute  the  cost  of  carpeting  each  of  the  rooms  with  Wilton 
carpet,  3/4  yd.  wide,  at  $1.25  per  yd.  Allow  9"  for  matching  design,  and 
allow  for  an  18"  border,  at  $1  35  per  yd.  Lay  the  carpet  in  the  direction 
that  will  be  more  economical. 


156 


BUSINESS   ARITHMETIC. 


Geometric  Conceptions — Solids. 
A  solid  is  that  which  has  three  dimensions,  length,  width 
and  thickness.     It  is  bounded  by  surfaces. 

A  polyhedron  is  a  solid  bounded  by 
plane  surfaces  called  faces.  A  poly- 
hedron having  two  parallel  and  equal 
bases,  and  three  or  more  sides  which 
are  parallelograms,  is  a  prism. 

A  prism  is  a  rectangular  solid  (A)  if 
it  is   bounded  by  six  rectangular  sur- 
faces.     It  is  a  cube  if  these    surfaces 
are  all  squares. 
A  pyramid   (B)   is  a  polyhedron  whose  faces,   with   one 
exception,  meet  in  a  common  point  called  the  vertex.     The 
faces  of  a  pyramid  are  therefore  triangles. 


A  cylinder  (C)  is  a  solid  bounded  by  two  eqUal  parallel 
circles  and  a  uniformly  curved  surface. 

A  conical  surface  is  a  surface  generated  by 
the  motion  of  a  straight  line  that  continu- 
ally intersects  a  fixed  curve,  frequently  a  cir- 
cle, and  passed  through  a  fixed  point.  A  cone 
(D)  is  a  solid  bounded  by  a  closed  conical  sur- 
face and  a  plane  surface. 

A  spherical  surface  is  a  curved  surface  every 


PRACTICAL   MEASUREMENTS. 


157 


point  of  which  is  equidistant  from 
a  point  called  the  center.  A  sphere 
(E)  is  a  solid  bounded  by  a  spher- 
ical surface.  A  hemisphere  is  one- 
half  of  a  sphere  cut  by  a  plane  pass- 
ing through  the  center. 


INTRODUCTORY   EXERCISE. 

A  cubic  foot  is  a  solid  1  foot  long,  1 
foot  wide  and  1  foot  thick.  How  many  cubic  feet  in  figure  (6)  if  each 
edge  is  1  foot  long?  How  many  cubic  feet  in  each  horizontal  row  of  (c)? 
Hovv'  many  rows?  How  many  cubic  feet  in  all?  How  many  rows  and 
units  per  row  in  one  vertical  slice  of  (d)?  How  many  sections?  How 
many  units  in  all  ?  What  is  the  area  of  the  end  face  ?  Of  the  top  2 
What  is  the  total  area  of  all  the  faces? 


It  is  evident  that  the  volume  of  a  rectangular  solid  is  equal 
to  the  product  of  its  three  dimensions.  These  dimensions 
must  ordinarily  be  expressed  in  units  of  the  same  denomi- 
nation. The  total  surface  area  is  equal,  evidently,  to  the 
sum  of  the  areas  of  the  individual  faces. 


ORAL    EXERCISE. 

Compute  the  cubic  contents,  or  volume,  of: 

1.  A  cube  of  6"  edge. 

2.  A  rectangular  solid  measuring  6'  by  5'  by  8'. 

3.  A  rectangular  solid  measuring  8'  by  4^'  by  10'. 

4.  An  excavation,  10'  by  20'  by  30'. 

5.  A  piece  of  metal  6"  wide,  12"  thick,  and  10'  long. 


158 


BUSINESS   ARITHMETIC. 


EXERCISE. 
1.    Compute  the  number  of  cubic  yards  of  earth  that  must  be  ex- 
cavated to  sink  this  cellar  to  a  depth  of  8'. 

Note.     Find    area    and    multiply    by 
32  FT.  depth.     Check,  by  dividing  into  separate 

prisms  as   shown  by  dotted  lines.     Why 
do  these  processes  bring  the  same  result? 

2.  Determine  the  number  of  cubic 
feet  in  a  stone  column  4^'  by  2'  by  18'. 

3.  Compute  the  cost  of  a  masonry 
wall,  40  long,  6'  high  and  2'  wide,  at  $4.36 
per  cu.  yd. 

4.  Determine  the  cubic  contents  of  a 
room  60'  by  25'  by  10'.  Determine  the 
surface  area  of  walls,  ceiling  and  floor. 

5.  A  school  room,  36'  by  30'  by  10', 
seats  42  people.  How  many  cubic  feet  of 
air  space  is  allowed  each  person? 

6.  At  the  rate  of  22  brick  per  cu.  ft., 
compute  the  cost  at  $2  per  M.  of  the 

bricks  necessary  for  a  wall  36'  long,  20'  high  and  24"  wide  for  half  the 
height  and  18"  wide  for  the  remaining  height. 


20  FT. 

u. 
vo 

12  FT. 


ORAL    EXERCISE. 

1.  Find  the  cubic  contents  of  this  rectangular  soUd? 
effect  on  the  volume  of  doubling  any  one  dimension? 

2.  If  the  volume  of  a  rectangular 
solid  is  180  cu.  in.,  and  the  product  of 
two  dimensions  is  30  sq.  in.,  what  is 
the  third  dimension? 

3.  If  two  dimensions  are  8'  and 
12',  what  is  the  third  dimension  of  a 
solid  having  a  volume  of  960  cu.  ft.? 

Note.  It  is  evident  that  the  third 
dimension  may  be  determined  by  divi- 
ding the  volume  by  the  product  of  the 
two  known  dimensions. 


What  is  the 


8  FT. 


EXERCISE. 

1.  8'  X  4.5'  X  ?  =  720  cu.  ft. 

2.  Select  two  dimensions  to  be  used  with  a  height  of  9',  to  determine 


PRACTICAL   MEASUREMENTS.  159 

a  rectangular  solid  of  120  cu.  ft.  of  volume.     How  many  solutions  are 
possible? 

3.  Name  three  dimensions  of  a  solid  of  a  cubic  content  of  5600  cu.  in. 

4.  Determine  the  height  of  a  room  30'  long  and  20'  6"  wide,  if  it  is  to 
have  a  cubic  content  of  5535  cu.  ft. 

The  Cylinder  and  Sphere. 

If  the  surface  of  a  paper  cylinder  is  cut  along  its  length, 
and  the  paper  rolled  back,  it  will  be  seen  to  have  the  form  of  a 
rectangle  whose  height  is  the  altitude  of  the  cylinder  and 
whose  base  is  the  length  of  the  circumference  of  the  base 
circle.  Evidently,  the  lateral  area  is  the  product  of  the  altitude 
by  the  circumference.  To  this  must  be  added,  to  determine 
the  entire  area,  the  areas  of  the  two  base  circles. 

ORAL    EXERCISE. 

1.  If  the  area  of  the  base  of  a  rectangular  solid  is  14  sq.  ft.,  what  is 
the  volume  per  foot  of  height? 

2.  If  the  area  of  the  circular  base  of  a  cylinder  is  30  sq.  in.,  what  is  the 
volume  per  foot  of  height?     What  is  the  volume  if  the  height  is  12  ft.? 

As  seems  evident  from  the  above,  the  volume  of  a  cylin- 
der equals  the  product  of  the  altitude  and  the  area  of  the  base. 

EXERCISE. 

1.  Allowing  for  1^"  overlap,  how  much  sheet  metal  is  required 
for  the  lateral  surface  of  a  hollow  cylinder  of  16'  altitude  and  3'  radius  of 
base? 

2.  Determine  the  cubic  contents  of  this  cylinder. 

3.  What  must  be  the  altitude  of  a  cylinder  of  4'  diameter  to  have  a 
cubic  content  of  600  cu.  ft.? 

4.  How  many  cu.  ft.  of  water  will  a  circular  cistern  of  12'  depth  and  5' 
radius  contain? 

5.  Determine  reasonable  dimensions  for  a  cistern  to  contain  5000  cu.  ft. 

6.  Metal  pieces  for  the  lateral  surfaces  of  cans  of  2"  radius  and  6" 
height  are  to  be  cut  from  sheet  metal  24"  by  32",  no  allowance  being  made 
for  overlap?     How  many  can  be  cut  from  each  sheet? 


160  BUSINESS  ARITHMETIC. 

The  surface  of  a  sphere  is  equal  to  ^  X  tt  X  ^^^  square  of  the 
radius;  and  the  volume  of  the  sphere  is  equal  to  4/3  X  ir  X  the 
cube  of  the  radius, 

EXERCISE. 

1.  Find  the  surfaces  of  spheres  of  radii  (a)  2";  (6)  6";  (c)  21''. 

2.  Find  the  volume  of  spheres  of  radii  (a)  5";  (6)  15";  (c)  26'  6". 

3.  Find  the  volume  of  a  hemisphere  of  .4'  6"  radius. 

4.  Find  the  radius  of  a  sphere  of  750  sq.  in.  of  surface. 

5.  How  many  square  inches  of  gilding  must  be  done  to  cover  a  sphere 
of  16"  radius? 

SPECIAL  APPLICATIONS. 

Cord  Measure. 

Fuel  wood,  tan  bark,  sometimes  stone,  etc.,  are  measured 
by  the  cord,  a  unit  pile  8'  long,  4'  wide  and  4'  high,  containing 
128  cu.  ft.  In  many  localities,  a  cord  of  fuel  is  a  pile  8'  long 
and  4'  high,  the  price  varying  with  the  width — as  half  sawed, 
quartered  sawed,  etc.  It  is  evident  that  if  a  pile  is  4'  high 
and  4'  wide,  the  number  of  cords  may  be  determined  from  the 
length,  as  measured  by  the  unit  length  of  8'.  Thus  a  36'  pile 
contains  4^  cords.  In  the  case  of  irregular  dimensions,  the 
measure  is  the  number  of  cubic  feet  in  a  cord. 

EXERCISE. 

1.  A  wood  lot  produces  piles  of  wood  of  regulation  4'  length  (width), 
of  dimensions  as  follows:  10'  long,  3'  high;  12'  long,  5'  high;  16'  long,  4' 
high;  20'  long,  4'  high.     How  many  cords  are  produced? 

2.  A  pile  of  hemlock  bark  measures  80'  by  4'  by  21'  and  contains 
how  many  cords? 

3.  Sixteen  acres  of  woodland  produce  a  pile  of  cord  wood  12'  high 
and  47'  long.     What  is  the  product  in  cords  per  acre? 

4.  Five  cords  of  wood,  regulation  size,  are  to  be  piled  in  a  shed  along 
a  wall  measuring  15'.     How  high  should  the  pile  be? 

Lumber. 

In  the  measurement  of  boards  of  less  than  1"  in  thickness, 
square  measure  of  surface  is  used.     In  the  measurement  of 


PRACTICAL   MEASUREMENTS. 


161 


boards  of  1"  thickness,  or  over,  such  as  heavy  boarding, 
planks,  joists,  scantUng,  and  heavy  dressed  timber,  the  unit 
measure  is  the  hoard  foot.  The  hoard  foot  is  a  hoard  1 '  long, 
12"  wide  and  1"  thick ,  or  its  equivalent.  Since  the  end  area 
is  12  sq.  in.,  the  board  foot  may  be  said  to  have  a  length  of  1' 
and  an  end  area  of  12  sq.  in. 


Board  Feet.  In  measuring  width,  except  in  the  more  valu- 
able woods,  the  next  lower  quarter  inch  of  width  is  sometimes 
taken.  Thus  a  6|"  width  is  reckoned  6f  and  a  ^\"  width, 
9'',  etc. 

In  stating  quantity  and  dimensions  of  an  order,  or  esti- 
mate, the  arrangement  is:  Number  of  pieces — thickness  in 
inches — width  in  inches — length  in  feet,  thus  30  pc,  6"  X  8" 
— 16'.    The  price  is  generally  stated  per  M  feet,  board  measure. 

In  computation,  both  on  paper  and  orally,  the  method  of 
cancellation  is  valuable. 

Illustration.  Compute  the  number  of  feet,  board  measure,  in 
36  pc,  2\"  X  10"  —16'. 

Solution.    3  2|  X  10  ^  no.  bd.  ft.  per  ft.   of 

=  1200,  no.  bd.  ft.  12  l^"^^^' 

This    no.  X  length  X  no.    pieces 
=  no.  bd.  ft. 


21X10X16X^6^ 
1 


Evidently  the  number  of  board  feet  is  the  product  of  the 
width  in  inches,  hy  the  thickness  in  inches,  hy  the  length  in  feet, 
by  the  number  of  pieces,  divided  hy  12.  The  quantity  to  cancel 
is  always  12.  Notice  that  this  is  one  of  the  rare  cases  where 
dimensions  are  not  expressed  in  the  same  unit. 
12 


162  BUSINESS   ARITHMETIC. 

ORAL    EXERCISE. 

Determine  the  number  of  board  feet  per  foot  of  length  in  timbers  of  the 
following  dimensions : 

1.  13"X4".        4.     2"X9".  7.     3"X6".  10.     3"X9". 

2.  8"X16".        5.     2"X8^  8.     2h"X&'.         11.     2"X3". 

3.  4"X12".        6.     4"X6".  9.     6"X9".  12.     2"X2". 
Determine  the  number  of  board  feet  in  these  timbers: 

13.  3"X4"— 16'.  16.  4"X8"— 12'. 

14.  4"X6"— 12'.  17.  2"X9"— 16'. 

15.  8"X10"— 6'.  18.  3"X6"— 12'. 
Determine  the  board  feet  in  these  items: 

19.     lOpc.  2"X6"— 10'.  20.    20pc.,  3"X4"— 12'. 

EXERCISE. 
Determine  the  number  of  feet,  board  measure,  in: 

1.  162  pc.  hemlock,  S"X10"— 32'. 

2.  40  pc.  white  pine,  2"  X 10"— 20'. 

3.  360  pc.  maple,  1|"X3"— 18'. 

4.  456pc.  oak,  3"X6"— 18'. 

At  $26  per  M.,  compute  the  cost  of: 

5.  150  joists,  2"X8"— 16'. 

6.  42  beams,  6"X9"— 24'. 

7.  78  scantling,  2"X4"— 16'. 

8.  How  much  timber  is  required  for  a  plain  board  fence  around  a  lot 
180'  by  246',  if  it  consists  of  two  strips  of  2"X4"  scantling  on  which  is 
nailed  1"  boarding  6'  high.  This  does  not  include  posts.  Allow  1/8 
for  waste  in  cutting. 

Capacity  of  Bins,  Tanks,  Etc. 

In  measuring  the  capacity  of  bins  and  tanks,  it  is  often 

necessary  to  express  the  common  dry  and  liquid  measures  in 

terms  of  cubic  feet. 

Illustrations.     (1)  Determine  the  capacity  in  gallons  of  a  rectangular 
iron  tank  measuring  4'  by  8'  by  3'. 
Solution.     (Abstract.) 
4'X8'X3' =96  cu.  ft.,  capacity. 
96  X 1728  =  capacity  in  cu.  in. 


PRACTICAL  MEASUREMENTS  163 

^^oqV^^  =^o.  of  gal.  (since  1  gal.  =231  cu.  in.)  =718+  gal.,  capacity. 

(2)  What  is  the  weight  of  the  water  the  tank  will  hold? 
Solution, 

96  cu.  ft.  =  contents. 

1  cu.  ft.  of  water  weighs  62^  lb. 

96X621  lb.  =6,000  lb.,  the  weight  of  water. 

EXERCISE. 

1.  Determine  the  capacity  in  bushels  of  a  rectangular  bin  measuring 
8'  by  4'  by  4'.  Determine  the  capacity,  also,  by  approximate  measure 
(1  bu.  =  li  cu.  ft.). 

2.  A  bin  in  a  certain  bam  can  be  allowed  a  floor  space  of  only  4'  by 
9'  6".  What  must  be  its  height  if  it  is  to  have  a  capacity  of  250  bu.? 
(Approximate  measure.) 

3.  Determine  reasonable  dimensions  for  a  bin  to  contain  400  bu.  of 
potatoes.     Use  1  c\i.  ft.  as  equivalent  to  .63  of  a  heaped  measure  bushel. 

4.  In  order  to  have  a  capacity  of  1,000,000  gal.,  a  reservoir  must  have 
a  capacity  of  how  many  cu.  ft.? 

5.  Compute  the  capacity,  in  gallons,  of: 

(a)  A  rectangular  tank,  4'  X 12'  X3'  6". 

(b)  A  cylindrical  tank  of  6'  diameter  and  8'  high. 

(c)  A  cylindrical  cistern  of  2'  6"  radius  and  16'  deep. 

(d)  A  vat  6'  square  and  4'  6"  high. 

6.  A  metal  tank,  known  to  have  a  capacity  of  60,000  gal.,  will  hold  for 
temporary  storage  how  many  bushels? 

7.  Construct  a  table  to  give  the  number  of  standard  gallons  in  volumes 
from  1  to  1000  cubic  feet. 

SPECIAL  INDIVIDUAL  PROBLEMS. 

1.  Estimate  the  quantity  and  cost  of  materials  for  a  cheap  board 
fence  to  enclose  a  vacant  lot  in  your  neighborhood.  Prepare  a  plot  of  the 
land  and  a  diagram  showing  character  of  fence. 

2.  Prepare  plans  and  estimates  for  a  modem  earth  croquet  court  (or 
tennis  court)  to  be  constructed  on  a  vacant  lot  in  your  neighborhood. 

3.  Report  on  the  total  number  of  square  feet  of  lighting  surface  in  the 
school  building.    Tabulate  results  (1)  by  rooms,  (2)  by  floors,  (3)  by  com- 


4.     Report  on  the  number  of  cubic  feet  of  air  space  in  the  building,  by 
rooms,  corridors  and  floors. 


CHAPTER  XXIV. 

MEASUREMENT   OF  TIME. 

1.     Longitude   and   Time. 

Longitude  and  time  are  matters  of  common  interest.     They 
are  also  fundamental  to  navigation,  astronomy  and  geography. 

Nortli  Pole 


Cape  Town 


South  Pole 


A  meridian  is  an  imaginary  line  on  the  surface  of  the  earth, 
extending  due  north  and  south  and  terminating  in  the  poles. 
It  is  therefore  a  semi-circumference  of  the  earth.  Any  me- 
ridian used  as  a  reference  Hne,  from  which  to  locate  places  on 

164 


MEASUREMENT  OF  TIME.  165 

the  earth's  surface  by  determining  their  distances  east  or  west 
of  it,  is  a  'prime  meridian.  It  is  now  the  common  custom  to 
use  the  meridian  through  the  Royal  Observatory  at  Green- 
wich, England,  as  a  prime  meridian. 

Distances  east  or  west  of  the  prime  meridian  are  termed 
respectively  east  longitude  and  west  longitude.  They  are 
measured  by  degrees,  minutes  and  seconds  of  circular  measure, 
as  shown  on  page  164,  where  the  meridians  intersect  the  equator. 
Each  place  has  its  own  meridian,  but  two  places  may  have  the 
same  meridian  even  when  thousands  of  miles  apart.  Longi- 
tude west  is  designated  by  the  symbol  '  +,'  and  longitude 
east  by  the  symbol  ^  — .'  Thus  the  longitude  of  New  York, 
which  is  west  of  Greenwich  is  +73°  58'  25.5'',  while  the  longi- 
tude of.  Brussels,  east  of  Greenwich,  is  —  4°  22'  9". 

From  a  study  of  the  diagram,  it  is  evident  that  the 
difference  in  longitude  of  two  places  may  be  obtained  by 
subtraction,  if  the  signs  of  the  individual  longitudes  are  the 
same;  and  by  addition  if  they  are  different. 

Since  the  earth  revolves  on  its  axis  once  in  twenty-four 
hours,  every  point  on  the  surface  revolves  in  a  circle,  or  passes 
through  360°  of  circular  measure,  in  that  time.  Hence  there 
is  a  close  relation  between  time  and  degrees  of  circular  measure. 

Since  360°  correspond  to  24  hours. 
1°  is  equivalent  to  1/360  of  24  hr.,  or  4  min. 
1'  is  equivalent  to  1/60  of  4  min.,  or  4  sec. 
1"  is  equivalent  to  1/60  of  4  sec,  or  1/15  sec. 

Since  24  hours  correspond  to  360°, 
1  hr.  is  equivalent  to  1/24  of  360°,  or  15°. 
1  min.  is  equivalent  to  1/60  of  15°,  or  15'. 
1  sec.  is  equivalent  to  1/60  of  15',  or  15". 

Regarding  the  sun  as  stationary  with  reference  to  the  earth, 
the  earth,  revolving  from  west  to  east,  carries  the  meridians 
under  the  sun  in  succession,  the  latter  seeming  to  move  toward 
the  west.  Since  all  meridians  pass  under  the  sun  in  suc- 
cession each  twenty-four  hours,  they  really  mark,  by  differ- 


166  BUSINESS  ARITHMETIC. 

ence  in  longitude,  the  difference  in  time  of  places  on  the  earth's 
surface.  In  science,  difference  of  longitude  is  often  expressed 
in  time.  Local  time  is  determined  at  each  place  by  calling 
the  instant  the  sun  crosses  its  meridian  12:00  noon. 

Illustration.     If  the  difiference  in  longitude  of  two  places  is  84°  30', 
what  is  the  difiference  in  time? 
Solution. 
84°  =  84  X 1/15  hr.     Since  1°  corresponds  to  1/15  hr. 

=  5fhr.  =5hr.  36min. 

30' =  30X4  sec.  =  2  Since  1'  corresponds  to  4  sec. 

84°  30'  =5  hr.  38  sec,  difiference  in  time. 

EXERCISE. 

1.  Do  points  in  the  east  or  west  of  a  country  receive  sunlight  first? 
Which  have  earlier  local  time? 

2.  If  two  places  have  different  west  longitudes,  which  has  earlier  time? 
Which  of  two  places  having  east  longitude  has  the  earlier  local  time? 

Evidently  the  more  western  points  have  the  later  time; 
the  more  eastern  the  earlier.  Since  time  bears  an  intimate 
relation  to  degrees  of  longitude,  it  is  a  simple  matter  to  com- 
pute one  from  the  other. 

Illustrations.     (1)  The  longitude  of  Albany  is  73°  44'  48",  and  of 
Ann  Arbor,  Mich.,  83°  43'  48".     Determine  their  difiference  in  time. 
Solution. 

83°  43'  48",  long,  of  Ann  Arbor, 
73°  44'  48",  long,  of  Albany, 
9°  59'  =  difiference  in  longitude. 

9X1/15  hr.  =3/5  hr.,  or  36  min. 
5dX4sec.     =236  sec,  or  3  min.  56  sec. 
9°  59'  =  39  min.  56  sec,  difif.  in  time. 

(2)  The  difiference  in  the  time  of  two  places  is  6  hr.  18  min.  12  sec 
Determine  the  difiference  in  longitude. 
Solution. 

6  hr.  18  min.  12  sec  =  difiference  in  time. 
6hr.      =  6X15°       =90° 
18  min.  =  18X15'        =  4°  30' 

12  sec  =12X15"       = 3' 

6  hr.  18  min.  12  sec.  =94°  33',  difiference  in  longitude. 


Athens, 

-23°  43'  55.5". 

Berlin, 

-13°  23' 43.5". 

Chicago, 

+87°  36'  42.0". 

New  York, 

,  +74°    0'    3.0". 

Paris, 

-  2°  20'  15.0". 

MEASUREMENT  OF  TIME.  167 

Note.  Both  these  solutions  are  based  directly  on  the  tables  of  com- 
parative values  given  on  page  165. 

Longitude  of  Well-Known  Places. 

Philadelphia,      +75°     9'  45.0". 
Rome,  -12°    27' 14.0". 

San  Francisco,    +122°  25' 40.8". 
Tokyo,  -139°  42' 30". 

EXERCISE. 

1.  If  the  difference  in  longitude  of  two  places  is  71°  18'  0",  what  is 
the  difference  in  time? 

2.  Compute  the  difference  in  local  time  between  (a)  Berlin  and  Phila- 
delphia; (6)  Chicago  and  San  Francisco;  (c)  Rome  and  Paris,  and  (d) 
Chicago  and  Tokyo. 

3.  A  man  coming  to  Chicago  found  that  his  watch,  set  by  local  time 
at  his  starting  point,  was  48  minutes  slow  by  local  Chicago  time.  Did 
he  come  from  east  or  west?     From  what  longitude? 

4.  When  it  is  5:00  P.  M.  at  Chicago,  it  is  what  time  at  (a)  San  Fran- 
cisco, (6)  Berlin,  (c)  Rome  and  (d)  New  York  ? 

5.  A  ship  sailing  in  longitude  183°  40'  9"  is  how  far  in  difference  of 
longitude  from  the  port  of  New  York.  Near  what  great  body  of  land  is 
the  ship? 

The  International  Date  Line  has  been  selected  as  the  180th 
meridian,  with  slight  changes  at  certain  points.  This  meridian 
in  the  Pacific  Ocean  passes  through  no  important  land  areas. 
Going  westward  ships  set  their  calendars  forward  one  day  at 
this  point;  going  east  they  set  them  back  one  day. 

Standard  Time.  If  every  place  used  its  strict  local  time,  it 
is  evident  that  endless  confusion  would  result  in  travel, 
especially  from  difficulties  in  constructing  time  tables.  On 
this  account  the  railways  of  the  United  States  and  Canada 
agreed  on  four  time  belts  for  the  two  countries.  The  belts 
are  approximately  15°  wide,  and  each  belt  takes  the  time  of 
its  center  meridian.  Thus  the  guide  meridians  are  15°,  or 
one  hour  apart.  The  belts  are  the  Eastern,  Central,  Moun- 
tain and  Pacific,  the  meridians  being  +75°,  +90°,  +105°  and 


168  BUSINESS  ARITHMETIC. 

+  120°.  Thus,  when  it  is  10:00  A.M.  in  any  town  in  the 
Eastern  time  belt,  it  is  9:00  A.M.  in  the  Central  belt,  8:00 
A.M.  in  the  Mountain  belt  and  7:00  A.M.  in  the  Pacific 
belt. 

Foreign  countries  are  adopting  similar  belts,  based  largely 
on  meridians  at  15°  intervals  from  Greenwich. 

2.     Commercial  Time. 

The  accurate  computation  of  time  between  dates,  and  of 
dates  of  maturity,  is  of  vital  importance  in  business.  All 
interest  charges  depend  on  time.  Many  business  and  com- 
mercial papers  run  for  fixed  times  to  dates  of  maturity. 
Considerable  merchandise  is  sold  on  limited  time.  Failure 
to  correctly  compute  a  payment  date  may  cause  a  business 
man  to  lose  standing  or  credit,  by  being  caught  unprepared  to 
meet  just  demands. 

Many  seemingly  minor  details  affect  time  and  date  com- 
putation. Time  may  be  computed  either  approximately 
or  by  exact  days,  and  dates  of  legal  maturity  are  affected  by 
varying  state  laws  affecting  holidays  and  Sundays  (see  page 
312). 

I.    Time  Between  Dates. 

Since  the  common  business  time  is  six  months,  or  less,  it  is 
possible  to  make  most  time  computations  without  use  of  paper. 
Where  paper  is  used,  the  method  is  often  different. 

(a)  Approximate  Time.     (Paper.) 

Illustration.  Find  the  time  between  June  21,  1908,  and  Jan.  17, 
1910. 

Solution.     Follow  the  method  of  denomi- 
nate numbers.     In  the  subtrahend,  1  mo.  is 
reduced  to  days,  making  30+17,  or  47  da.; 
1  yr.     6  mo.    26  da.         and  1  yr.  is  reduced  to  12  mo. 

Note.  In  reduction,  one  month  is  considered  30  days,  but  if  a  given 
date  contains  the  thirty-first  day  it  is  retained. 


Yr. 

Mo. 

Da. 

1910 

1 

17 

1908 

6 

21 

MEASUREMENT   OF  TIME.  169 

EXERCISE. 

Find  the  time  between: 

1.  Jan.    27,  1906,  and  Aug.  29,  1909.        4.     3/    3/  '07  and  6/  I/'IO. 

2.  Oct.    21,  1907,  and  Dec.  15,  1912.        5.     7/    5/  '06  and  10/  17/  '06. 

3.  Jan.   31,  1908,  and  Sept.  12,  1909.        6.     3/  21/  '07  and  5/  18/  '09. 

7.  Money  borrowed  Apr.  16, 1907,  and  due  to-day,  has  drawn  interest 
for  how  long? 

8.  I  am  given  until  Oct.  1  to  pay  a  debt  due  Feb.  11.  For  how  long  is 
time  of  payment  extended? 

(b)  Approximate  Time.     (Mental.) 

Illustrations.  (1)  Find  the  time  between  Mar.  12,  1906,  and  May 
17,  1908. 

Solution.  Reason  as  follows:  From  Mar.  12,  1906,  to  Mar.  12,  1908, 
2  years;  from  Mar.  12,  1908.  to  May  12,  2  months;  from  May  12  to  May 
17,  5  days.     Total  time,  2  yr.  2  mo.  5  da. 

Note.  The  process  involves  adding  to  the  earlier  date,  on  the  principle 
used  in  making  change. 

(2)  Find  the  time  between  Aug.  28,  1913,  and  July  12,  1914. 

Solution.  Aug.  28  (8th  mo.)  to  June  28  (6th  mo.)  =  10  mo.  June  28 
to  July  12  =  2  +  12  da.  =  14  da.     Total  time,  10  mo.  14  da. 

EXERCISE. 

Compute  the  time  between : 

1.  Feb.  11  and  Apr.  29.  5.  Mar.  16,  1912,  and  Dec.  15,  1913. 

2.  June  16  and  Dec.  31.  6.  Feb.  28,  1912,  and  Nov.  29,  1912. 

3.  May  31  and  Nov.  24.  7.  Aug.  13  and  to-day. 

4.  Mar.  17  and  Oct.   9.  8.  May  19  and  to-day. 

Exact  Time.  In  computing  by  exact  time,  the  true  number 
of  days  in  each  month  is  taken  into  account.  Usually  only 
one  of  the  extreme  days  is  included,  although  some  bankers 
include  both.  Exact  time  is  computed  for  periods  less  than 
a  year.  Longer  terms  are  reckoned  in  calendar  years  and 
exact  days. 

(c)  Exact  Time.     (By  Table.) 

The  following  table  shows  the  calendar  year  and  the  day  of  the  year  of 
each  calendar  day.  Thus  Feb.  17  is  the  48th  day  of  the  year.  Other 
forms  of  tables  are  very  common. 


170 


BUSINESS   ARITHMETIC. 


Table 

FOR  THE 

Calculation  of  Time 

Jan.  1 

1 

Feb.  1 

32 

Mar.  1  60 

Apr.  1 

91 

May  1 

121 

June  1 

152 

2 

2 

2 

33 

2  61 

2 

92 

2 

122 

2 

153 

3 

3 

3 

34 

3  62 

3 

93 

3 

123 

3 

154 

4 

4 

4 

35 

4  63 

4 

94 

4 

124 

4 

155 

5 

5 

5 

36 

5 

64 

5 

95 

5 

125 

5 

156 

6 

6 

6 

37 

6 

65 

6 

96 

6 

126 

6 

157 

7 

7 

7 

38 

7 

66 

7 

97 

7 

127 

7 

158 

8 

8 

8 

39 

8 

67 

8 

98 

8 

128 

8 

159 

9 

9 

9 

40 

9 

68 

9 

99 

9 

129 

9 

160 

10 

10 

10 

41 

10 

69 

10 

100 

10 

130 

10 

161 

11 

11 

11 

42 

11 

70 

11 

101 

11 

131 

11 

162 

12 

12 

12 

43 

12 

71 

12 

102 

12 

132 

12 

163 

13 

13 

13 

44 

13 

72 

13 

103 

13 

133 

13 

164 

14 

14 

14 

45 

14 

73 

14 

104 

14 

134 

14 

165 

15 

15 

15 

46 

15 

74 

15 

105 

15 

135 

15 

166 

16 

16 

16 

47 

16 

75 

16 

106 

16 

136 

16 

167 

17 

17 

17 

48 

17 

76 

17 

107 

17 

137 

17 

168 

18 

18 

18 

49 

18 

77 

*  18 

108 

18 

138 

18 

169 

19 

19 

19 

50 

19 

78 

19 

109 

19 

139 

19 

170 

20 

20 

20 

51 

20 

79 

20 

110 

20 

140 

20 

171 

21 

21 

21 

52 

21 

80 

21 

111 

21 

141 

21 

172 

22 

22 

22 

53 

22 

81 

22 

112 

22 

142 

22 

173 

23 

23 

23 

54 

23 

82 

23 

113 

23 

143 

23 

174 

24 

24 

24 

55 

24 

83 

24 

114 

24 

144 

24 

175 

25 

25 

25 

56 

25 

84 

25 

115 

25 

145 

25 

176 

26 

26 

26 

57 

26 

85 

26 

116 

26 

146 

26 

177 

27 

27 

27 

58 

27 

86 

27 

117 

27 

147 

^  27 

178 

28 

28 

28 

59 

28 

87 

28 

118 

28 

148 

28 

179 

29 

29 

— 

— 

29 

88 

29 

119 

29 

149 

29 

180 

30 

30 

— 

— 

30 

89 

30 

120 

30 

150 

30 

181 

31 

31 

— 

— 

31 

90 

— 

— 

31 

151 

Illustration.     (1)  The  time  from  Aug.  17  to  Dec.  21  is  ?  days. 
Solution.     From  table.     Dec.  21=  355th  da. 
Aug.  17  =  229th  da. 
The  difference  equals  elapsed  time  =126  da. 
(2)  Nov.  11  to  Jan.  26=  ?  da. 

Solution  (a).     Nov.  15  is  315th  day.    365  da. -315  da.  =  50  da. 
Jan.   26  is  26th  day  of  next  year  26 

Total  time  76  da. 

Solution  (6).     The  complementary  time  between  Nov.  11  and  Jan.  26 
is  the  time  between  Jan.  26  and  Nov.  11. 

Nov.  11=  315th  da. 
Jan.   26=  26th  da. 
Complementary  elapsed  time  =289  da. 

True  elapsed  time =365  da. -289  da.  =  76  da. 


MEASUREMENT   OF  TIME. 


171 


Table  for  the  Calculation  of  Time. 


July 


182 

Aug.  1 

213 

183 

2 

214 

184 

3 

215 

185 

4 

216 

186 

5 

217 

187 

6 

218 

188 

7 

219 

189 

8 

220 

190 

9 

221 

191 

10 

222 

192 

11 

223 

193 

12 

224 

194 

13 

225 

195 

14 

226 

196 

15 

227 

197 

16 

228 

198 

17 

229 

199 

18 

230 

200 

19 

231 

201 

20 

232 

202 

21 

233 

203 

22 

234 

204 

23 

235 

205 

24 

236 

206 

25 

237 

207 

26 

238 

208 

27 

239 

209 

28 

240 

210 

29 

241 

211 

30 

242 

212 

31 

243 

Sept. 


1 
2 
3 
4 

6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 


244 

Oct.  1 

274 

Nov.  1 

305 

Dec.  1 

245 

2 

275 

2 

306 

2 

246 

3 

276 

3 

307 

3 

247 

4 

277 

4 

308 

4 

248 

5 

278 

5 

309 

5 

249 

6 

279 

6 

310 

6 

250 

7 

280 

7 

311 

7 

251 

8 

281 

8 

312 

8 

252 

9 

282 

9 

313 

9 

253 

10 

283 

10 

314 

10 

254 

11 

284 

11 

315 

11 

255 

12 

285 

12 

316 

12 

256 

13 

286 

13 

317 

13 

257 

14 

287 

14 

318 

14 

258 

15 

288 

15 

319 

15 

259 

16 

289 

16 

320 

16 

260 

17 

290 

17 

321 

17 

261 

18 

291 

18 

322 

18 

262 

19 

292 

19 

323 

19 

263 

20 

293 

20 

324 

20 

264 

21 

294 

21 

325 

21 

265 

22 

295 

22 

326 

22 

266 

23 

296 

23 

327 

23 

267 

24 

297 

24 

328 

24 

268 

25 

298 

25 

329 

25 

269 

26 

299 

26 

330 

26 

270 

27 

300 

27 

331 

27 

271 

28 

301 

28 

332 

28 

272 

29 

302 

29 

333 

29 

273 

30 

303 

30 

334 

30 

— 

31 

304 

— 

— 

31 

335 
336 
337 
338 
339 
340 
341 
342 
343 
344 
345 
346 
347 
348 
349 
350 
351 
352 
353 
354 
355 
356 
357 
358 
359 
360 
361 
362 
363 
364 
365 


EXERCISE. 


Compute  the  time  between ; 

1.  Feb.  11  and  Oct.  16. 

2.  Mar.  19  and  Nov.  10. 

3.  Jan.  15  and  Oct.  29. 

4.  Apr.  24  and  Dec.  27. 

5.  Feb.  28  and  Aug.  7. 


6.  July  15  and  Oct.  1. 

7.  Mar.  27,  1910,  and  Oct.  16,  1911. 

8.  Apr.  14,  1909,  and  Nov.  22,  1911. 

9.  May  29,  1910,  and  Dec.  19,  1911. 
10.  Nov.  16,  1912,  and  Apr.  26,  1914. 


11.    Show  how  to  use  the  table  in  leap  years. 

(d)  Exact  Time.     (Without  Tables.) 
Illustrations.     Find  the  time  between  Aug.  18  and  Oct.  27. 


172  BUSINESS   ARITHMETIC. 

Solution  (a).    Addition  method.     Aug.  18  to  31  =  13  da. 

Sept.    1  to  30  =  30 

Oct.     1  to    7  =  27 

Total  time       =70  da. 
Note.     Mentally,  each  group  is  added  to  the  previous  total,  as  found. 
One  would  say  13,  43,  70. 

Solution  (&).     From  approximate  time. 

Reason  as  follows:  Aug.  18  to  Oct.  27,  2  mo.  9  da.,  or  69  da.  But  this 
period  includes  a  31st  day  in  August.  Therefore  the  exact  time  is  69  da. 
+  1  da.,  or  70  da. 

Note.  Since  February  has  28  days,  this  method  involves  subtraction 
when  the  period  includes  the  last  of  this  month. 

EXERCISE. 

Find  the  exact  number  of  days  from : 

1.  Aug.  28  to  Dec.  29.  6.  Jan.  5  to  Mar.  8. 

2.  Sept.  19,  1907,  to  Jan.  25,  1908.  7.  Nov.  11  to  date. 

3.  Oct.  1  to  Jan.  13.  8.  Oct.  16  to  date. 

4.  Aug.  12  to  Dec.  3.  9.  Feb.  21  to  June  9. 

5.  Mar.  28  to  Apr.  14.         10.  Mar.  3  to  Aug.  2. 

II.    Dates  of  Matukity. 

In  computing  final  date?  and  dates  of  maturity,  it  is  neces- 
sary to  know  the  initial  date  and  the  "term"  or  period  of  time. 
If  the  term  is  stated  in  months  and  days,  approximate  methods 
are  used;  if  stated  in  days,  as  "90  days,"  exact  methods  are 
used.     "Day"  terms  are  seldom  over  90  days. 

(a)  Approximate  Method. 

Illustration.  A  payment  due  1  yr.  7  mo.  19  da.  from  Aug.  11, 1912, 
is  due  on  what  date? 

Solution.     (Paper.)     Write  in  denominate  form.      1912      8      11 
(Add  and  reduce  upward.)  1      7      19 

1914      3      30 
Date  due,  Mar.  30,  1914. 
Solution.     (Mental.) 

Reason  as  follows:  Aug.  11,  1912,  +  1  yr.=Aug.  11,  1913;  Aug.  11, 
1913,+7  mo.  =  Mar.  11,  1914;  Mar.  11  +  19  da.  =  Mar.  30,  1914. 


MEASUREMENT   OF  TIME.  173 

EXERCISE. 

Find  the  final  date. 

Initial  Date.  Term. 

1.  Jan.   27,  1907  1  yr.  3  mo.  18  da. 

2.  Feb.  16,  1913  2  yr.  5  mo.  11  da. 

3.  Mar.    9,  1908  1  yr.  11  mo.     3  da. 

4.  Apr.   14,  1908  2  yr.  5  mo.  29  da. 

5.  May  30,  1908  1  yr.  2  mo.  11  da. 

6.  Jmie  14,  1912  9  mo.  19  da. 

7.  Aug.  21,  1908  1  yr.  5  mo.  25  da. 

8.  Sept.    4,  1913  2  yr.  5  mo.     6  da. 

ORAL    EXERCISE. 

1.  What  date  is  1   mo.  from  Jan.  28?     From  Jan.  29?    Jan.  30? 
Feb.  1?     Mar.  30?     Apr.  6?    June  20?    Aug.  31? 

Note.     If  there  is  no  corresponding  date,  the  last  date  in  the  month 
is  taken.     Thus,  1  mo.  from  Oct.  31  is  Nov.  30. 

2.  Mar.    6+lmo.  =  ?  •        6.  Nov.  30+3  mo.  =? 

3.  Apr.  20+3  mo.  =?  7.  Dec.  31+2  mo.  =  ? 

4.  June  30+4  mo.=?  8.  Dec.  31+6  mo.  =  ? 

5.  July  31+4  mo.  =?  9.  July  14+8  mo.  =  ? 

(b)  Exact  Method.     {By  Table,  pp.  170,  171.) 

ORAL    EXERCISE. 

1.  What  day  of  the  year  is  June  14?    What  day  of  the  year  is  60  days 
later?     What  date? 

2.  Find  a  date  90  days  in  advance  of  June  29. 
{Solution.    June  29  =  180th  day.     180  da.  +90  da.  =270  da. 

By  table,  the  270th  day  is  Sept.  27.) 
Find  the  final  dates. 


Initial  Date. 

Term. 

Initial  Date. 

Term. 

3.     June  26 

•    90  da. 

7. 

Oct.   14 

125  da. 

4.     Aug.  31 

90  da. 

8. 

June  11 

45  da. 

5.     Jan.    30 

30  da. 

9. 

July  29 

40  da. 

6.     Feb.  26 

60  da. 

10. 

Aug.  27 

165  da. 

174  BUSINESS  ARITHMETIC. 

(c)  Exact  (Periodic)  Method. 

ORAL   EXERCISE. 

1.  What  date  comes  30  days  after  Jan.  31?  After  Mar.  31?  After 
April  20?     After  July  16? 

2.  When,  for  periods  of  30  days,  is  the  final  date  the  corresponding  date 
of  the  proper  month?     When  is  it  earlier?     When  later? 

3.  Find  the  date  of  maturity  of  a  paper  running  90  days  and  dated 
Jan.  26. 

Solution.  Jan.  26+30  da.  =  Feb.  25,  since  Jan.  has  31  da.;  Feb.  25 
+30  da.  =  Mar.  27.  (Why?)  Mar.  27+30  da.  =Apr.  26.  Therefore,  the 
paper  is  due  Apr.  26. 


Find  the  dates  of  maturity. 
Initial  Date.     Term. 

Initial  Date. 

Term. 

4.     Jan.   29 

60  da. 

8.     June  27 

90  da. 

5.     Feb.   17 

90  da. 

9.     Aug.  14 

75  da. 

6.     Apr.  30 

60  da. 

10.     Sept.  19 

60  da. 

7.     May  19 

40  da. 

11.     Nov.  30 

60  da. 

INDIVIDUAL  ORIGINAL  WORK. 

1.  Report  on  "The  Arithmetic  of  Time  Table  Construction."  Give 
illustrations.  Give  some  idea  of  the  character  of  information  obtainable 
from  time  tables. 

2.  Prepare  a  time  table  and  car  schedule  for  a  double  track  street 
railway.  Estimate  the  number  of  motors,  trailers,  and  the  working  force 
necessary  for  efficient  service.  Length  of  trip  40  minutes  from  6  A.M. 
to  9  P.M.;  35  minutes  from  9  P.M.  to  6  A.M.  3  minute  schedule,  7  to 
10  A.M.;  3  to  7  P.M.;  5  minute  schedule,  6  to  7  A.M.;  10  A.M.  to  3  P.M.; 
7  to  10  P.M.  8  minute  schedule,' 10  to  12  P.M.  30  minute  schedule,  12 
P.M.  to  6  A.M. 

3.  You  are  to  leave  Washington  on  April  30  for  a  tour  of  the  following 
places  to  look  after  business  interests  for  me:  Boston,  Portland,  Me., 
Albany,  Buffalo,  New  York,  Cincinnati,  Memphis,  St.  Louis,  New  Orleans? 
Charleston,  S.  C,  Charlestown,  W.  Va.,  and  Minneapolis.  Examine 
railway  time  tables  and  submit  detailed  report  showng  routes  selected  ^ 
provisional  time  table,  etc.  If  possible  make  an  estimate  of  expense. 
Plan  for  at  least  ten  business  hours  in  Boston,  New  York,  Cincinnati, 
and  St.  Louis,  eight  hours  in  Minneapolis,  and  at  least  four  hours  in  each 
of  the  other  places. 


BOILING  POINT 


OF  WATER       +80° 


CHAPTER   XXV. 
PRACTICAL  MEASUREMENTS— Coniinwed 

1.     Measure  of  Temperature. 

Ordinary  temperatures  are  measured  by  thermometers, 
graduated  on  one  of  three  scales:  (1)  the  Fahrenheit,  commonly 
used  in  business  and  private  life;  (2)  the  Centigrade ,  used  in 
scientific  work,  and  in  countries  employing  the  metric  system 

and  (3)  the  Reaumur,  used  on  the 

continent  of  Europe,  especially  in 

Germany. 

The  common  thermometer,  con- 
sisting of  its  column  of  mercury  in 
a  glass  tube,  cannot  be  used  for 
extreme  high  temperatures.  For 
such  temperatures,  especially  those 
— .  over  500°,  instruments  of  many 
varied  type5  are  constructed.  Sev- 
eral, for  example,  measure  tem- 
perature by  the  relative  expansion 
of  two  metals,  such  as  copper  and  iron. 

The  level  at  which  the  mercury  stands,  when  the  thermo- 
meter is  in  contact  with  melting  ice  is  marked  0°  on  the  Cen- 
tigrade and  Reaumur  scales,  and '+32°  on  the  Fahrenheit 
scale.  When  in  contact  with  water  commencing  to  boil,  the 
mercury  level  is  marked  +80°,  Reaumur;  +100°,  Centigrade; 
and  +212°,  Fahrenheit.  These  limits  being  determined,  the 
degrees  of  the  scale  are  uniformly  marked  off.  Degrees  below 
0°  are  marked  "  -,"  those  above  "  +." 


t 


ORAL    EXERCISE. 
1.     Name  the  number  of  degrees  between  freezing  and  boiling  point 
on  each  scale. 

175 


176 


BUSINESS   ARITHMETIC. 


2.  Rank  the  individual  degrees  in  order  of  size. 

3.  1°  R.  is  what  part  of  1°  F.?     Of  V  C? 

4.  rF.iswhatpartofrC?    OfTR.? 
6.  1°  C.  is  what  part  of  1°  R.?    Of  1°  F.? 

Scale  Reduction. 

Owing  to  the  lack  of  uniformity  on  the  use  of  scales,  the 
problem  of  reduction  from  one  to  another  is  very  common. 


Illustrations.     (1)  40''F.  =  ?° 

c. 

(2)  12°  F.  =  ?°C. 

Solution. 

Solution. 

'.'  180°  F.  =  100°  C,  between  limits. 

32° -12°  =  20°  F.,  distance  below 

rF.-^^^g^Jc.,or5/9°C. 

0°C. 
20°  F.  = -20X5/9°  C. 

40°  F.  =  40° -32°  above  zero, 

or  8° 

= -11.11°  C. 

above  zero. 

(4)   -5°C.  =  ?°F. 

8  X  5/9°  =     4.44  +  °  C. 

Solution. 

Solution. 

5°C.  =  5X9/5°F.=  9°F. 

'.-  100°  C.  =  180°  F. 

But  0°C.=32°F. 

1°  C.  =  9/5°  F. 

.-.  -5°C.=32°-9°F.=23°F. 

25°  C.  =25x9/5°  F. ; 

=  45°  F.,  above  freezing. 
But  0°  C.  =  32°  F. 
.'.  25°  C.  =  45°  F.  +  32°  F.  =  77°  F. 

EXERCISE. 
Express  each  of  the  following  temperature  reading  on  two  other  scales: 

1.  +196°  F.       5.     +18.6°  F.         9.     +182°  C.       13.     +67.2°  R. 

2.  +108°  F.       6.     -18°  C.         10.     +46.5°  C.      14.     -50°  R. 

3.  +64°  F.         7.     +64°  C.         11.     +18°  R.        15.     +112°  R. 

4.  -13°  F.         8.     -30°  F.  12.     -66°  C.        16.     +100**R. 
17.    Express  the  following  table  in  Centigrade  scale: 

Approved  Cold  Storage  Temperatures. 
Degrees  Fahrenheit. 


Beef 36  to  40. 

Pork 29  to  32. 

Lamb,  mutton 32  to  36. 

Fish 25  to  28. 

Oysters 33  to  45. 

Fruits 32  to  36. 


Poultry,  frozen 25  to  30. 

Poultry,  to  freeze 5  to  10. 

Eggs 32  to  34. 

Butter 32  to  38. 

Vegetables 34  to  40. 

Canned  goods 38  to  40. 


PRACTICAL   MEASUREMENTS.  177 

18.  Find  the  missing  temperatures  in  these  tables: 

(a)  Boiling  Points.  (6)  Melting  Points. 

Substance.                     C.        F.  Substance.  C.         F. 

Benzine 176"         Copper 2012° 

Chloroform 140"        Platinum 1775"  

Mercury 676"        Sulphur 115" 

Sulphur 570"        Zinc 779° 

19.  Construct  a  conversion  table,  Centigrade-Fahrenheit,  for  0"  to  50° 
Centigrade. 

2.  Composite  Units  of  Measure. 
It  sometimes  happens  that  an  accurate  measure  of  values 
cannot  be  made  by  a  simple  standard  unit,  because  of  some 
special  influencing  condition.  Thus  the  freight  traffic  of  a 
railway  cannot  be  measured  accurately  by  tons,  because  some 
tons  are  carried  only  short  distances,  while  others  are  carried 
great  distances;  while  if  measured  by  freight  earnings,  the 
factor  of  quantity  is  lost  sight  of.  To  meet  the  difficulty,  a 
composite  unit,  the  ton-mile,  was  adopted.  This  unit  is 
one  ton  carried  one  mile.  If  the  ton  is  carried  ten  miles  the 
traffic  amounts  to  10  ton-miles;  if  5  tons  are  carried  8  miles, 
the  traffic  amounts  to  5X8,  or  40  ton-miles.  Measured 
by  such  a  unit  the  full  effect  of  both  quantity  and  distance 
is  taken  into  account.  Moreover,  it  makes  possible  a  com- 
parison of  the  traffic  of  different  branches,  or  of  independent 
roads.  Composite  measures  of  a  similar  order  are  increasingly 
used  in  science  and  in  statistics. 

A  Few  Other  Composite  Unit  Measures. 

Unit  of  work.     A  foot-pound — the  work  done  in  raising  one 
■pound  one  foot. 

Unit  of  power.     The  horse-power.     It  is  equal  to  33,000 
foot-pounds  per  minute,  or  550  foot-pounds  per  second. 

The  irrigation  unit.     The  acre-foot.     The  quantity  of  water 
required  to  cover  one  acre  to  a  depth  of  one  foot. 
13 


178  BUSINESS   ARITHMETIC. 

The,  unit  of  freight  traffic.  The  ton-mile.  The  equivalent  of 
one  ton  of  freight  carried  one  mile. 

The  unit  of  passenger  traffic.  The  passenger-mile.  One 
passenger  carried  one  mile. 

The  unit  of  train  traffic.  The  train-mile.  One  train  run  one 
mile. 

Illustration.  What  horse-power  in  an  engine  will  raise  4,950,000  lb. 
10  ft.  in  5  min.? 

Solution. 

4,950,000X10  =  49,500,000,  no.  of  ft.  p. 
49,500,000  4- 5  =  9,900,000,  ft.  p.  per  min. 
9,900,000^33,000=300,  no.  of  horse  power. 

ORAL   EXERCISE. 

1.  How  many  foot-pounds  of  work  are  performed  in  raising  48  lb. 
15  ft.? 

2.  How  many  acre-feet  of  water  are  required  to  supply  a  2^  ft. 
depth  to  an  area  of  4^ A.? 

3.  What  is  the  ton-mileage  of  a  car  load  of  60  T.,  carried  a  distance 
of  56  mi.? 

4.  Compute  the  passenger  mileage  of  32  people,  carried  an  average 
distance  of  15  mi. 

EXERCISE. 

1.  Compute  the  horse-power  employed  in  raising  30,000  lb.  an  average 
distance  of  48  ft.  per  min. 

2.  Compute  the  horse-power  performed  in  raising  5000  gal.  of  water 
per  sec,  a  distance  of  10  ft. 

Note.    Express  the  water  in  terms  of  weight. 

3.  The  owner  of  the  tract  of  land  shown  on  page  149  contracts  for  a 
supply  of  water  of  2^  ft.  per  annum,  for  irrigation.  What  will  it  cost 
him,  at  $46  per  acre-foot? 

4.  A  storage  reservoir  is  to  be  built  with  a  capacity  of  50,000  acre-feet. 
What  must  be  its  capacity  in  gallons? 

5.  A  metal  casting  weighing  48  T.  150  lb.  is  carried  a  distance  of  169.3 
mi.     Compute  the  ton-mileage. 

6.  In  a  recent  year,  the  railways  of  the  country  carried  1,309,899,165 
T.  of  freight  an  average  distance  of  133.23  mi.     Compute  the  ton-mileage. 


PRACTICAL  MEASUREMENTS.  179 

7.  In  the  year  above  mentioned,  715,419,682  passengers  were  carried  a 
passenger  mileage  of  21,923,213,536.  What  was  the  average  distance 
each  passenger  was  carried? 

8.  In  the  year  above  referrred  to  the  passenger  train  mileage  was 
440,464,866  train  miles.     What  was  the  passenger  mileage  per  train-mile? 

3.    The  Formula. 

In  estimating  of  every  kind,  it  frequently  occurs  that  a 
general  rule  for  the  solution  of  problems  of  a  certain  type  is 
reduced  in  statement  to  a  simple  form  of  an  equation,  em- 
ploying words,  or  more  frequently,  symbols. 

Illustration.  The  fact  that  the  area  of  a  triangle  is  equal  to  one-half 
the  product  of  base  and  altitude  may  be  expressed: 

.          r     X  •       1       Altitude  X  base 
Area  of  a  triangle  =  ^ . 

We  may  need,  however,  to  compute  areas  of  many  different  figures. 
In  such  a  case,  we  may  let  "A"  stand  for  the  area  of  any  figure,  "a"  for 
altitude,  "6"  for  base,  "c"  for  circumference  and  "i2"  for  radius,  the 
Greek  letter  "tt"  for  the  ratio  of  diameter  to  circumference.  Using 
symbols,  certain  common  rules  for  area  reduce  to  the  following: 

1.  For  area  of  a  triangle,  A  =  —^  ,  or  simply  -^. 

2.  For  area  of  a  rectangle,  A=ab. 

3.  For  area  of  a  circle,  A  =  irE^. 

Frequently  the  determination  of  the  formula  is  a  matter  of 
summing  up  many  experiments  (in  arbitrary  rules — called 
empirical  rules),  or  of  complicated  computations  by  higher 
mathematics.  But  once  formulated,  computations  may  often 
be  made  by  simple  arithmetic  if  in  any  problem  numerical 
values  are  known  for  all  except  one  symbol. 

Illustrations:  (1)  Find  the  area  of  a  circle  of  6"  radius. 
Solution. 

Formula,     A  =  ttB^. 

But  TT =3.1416,      R  =  6,     fi»  =  36. 

Substituting  in  the  formula, 
A  =3.1416X36. 
=  113.0976  sq.  in. 


180  '  BUSINESS  ARITHMETIC. 

(2)  Find  the  base  of  a  rectangle  whose  area  is  135  sq.  in.  and  whose 
altitude  is  9". 

Solution. 

Formula,      A=ab. 
But  A  =  135and6=9. 

Substituting  in  the  formula, 
135  =  9a. 
15  =  a,  the  altitude.     (Dividing  by  9.) 

Note.  Care  should  be  taken  that  denominations  are  correct.  Thus, 
in  above  case,  if  area  had  been  stated  in  square  feet  while  other  factors 
were  in  inches,  both  factors  should  have  been  reduced  to  a  common  de- 
nomination of  feet  or  inches. 


EXERCISE. 

1.  Express  in  symbols  the  value  of  the  circumference  of  a  circle,  in 
terms  of  the  radius  and  constant  ratio. 

2.  Express  in  symbolic  form  (a)  the  area  of  a  parallelogram,  (6)  the 
surface  of  a  cube,  (c)  the  volume  of  a  rectangular  solid,  (d)  the  surface  area 
of  such  a  solid,  (e)  the  perimeter  of  a  parallelogram. 

3.  The  safe  weight  of  a  floor  for  a  certain  type  of  bridge  is  80  lb. 
per  sq.  ft,     Express  this  fact  in  formula  for  any  sized  bridge  of  the  type. 

4.  Find  by  the  bridge  formula,  the  safe  load  of  a  bridge  90  ft.  long 
and  17  ft.  6  in.  wide. 

xb 

5.  If  a:  =  12,  6  =  6  and  c  =  5,  find  A,  in  the  formula  A-  —  . 

c 

6.  Find  S,  in  the  formula  S  =  iTrR^,  if  R  =  17. 

7.  The  number  of  board  feet  in  a  rectangular  timber  =  length  in  feet 
X  breadth  in  feet  X  thickness  in  inches. 

a.     Express  this  formula  in  symbols. 

h.    Use  the  formula  to  determine  the  number  of  board  feet  in  a 
timber  12'  long,  9"  wide  and  2"  thick. 

8.  The  pressure  of  wind  blowing  directly  against  an  object  is  sometimes 
measured  by  the  formula  P  =  .0023  7^  in  which  P= pressure  in  pounds 
per  sq.  ft.,  and  7  =  velocity  in  feet  per  second.  Find  the  pressure  per 
square  foot  under  a  wind  blowing  40  miles  per  hour. 

9.  A  roof  is  so  inclined  that  the  wind  pressure  is  .45  of  what  it  would  be 
if  the  wind  struck  it  perpendicularly.  What  is  the  total  wind  pressure  on 
the  roof,  which  measures  18'  by  60',  when  the  wind  is  blowing  32  miles 
per  hour? 


PRACTICAL   MEASUREMENTS.  181 

10.     The  volume  of  boards  that  may  be  sawed  from  a  round  log  may  be 
computed  by  the  formula 

L{d-^)dl2 


V  = 


8 


in  which  d  =  diameter  in  inches,  L  =  length  in  feet,  and  y= volume  in  feet, 
board  measure.  Compute  the  board  measure  in  a  60  ft.  log  of  15  in. 
diameter. 


CHAPTER  XXVI. 

THE   METRIC  SYSTEM, 
i 

The  metric  system  is  a  decimal  system  of  measures  adopted 
by  the  French  government  about  1800.  It  is  used,  scien- 
tifically, the  world  over,  and  commercially  in  many  civilized 
countries.  It  is  authorized  in  the  United  States  and  Great 
Britain,  but  is  used  only  in  a  part  of  their  foreign  trade. 

The  system  is  based  on  the  meters  a  unit  of  length  of  approxi- 
mately 39.37  in.,  which  was  assumed  to  be  exactly  1/10,000,000 
of  the  distance  from  equator  to  pole.  Later  measurements, 
however,  have  proven  this  computation  incorrect.  The  unit 
of  capacity,  the  liter,  is  a  cube  of  -^^  m.  edge.  The  unit  of 
weight,  the  gram,  is  the  weight  of  a  cube  of  water  of  1/100  m. 
edge.  The  tables  contain  multiples  and  decimals  of  units. 
Multiples  are  known  by  the  Greek  prefixes:  myria  (10,000), 
kilo  (1000),  hekto  (100)  and  deka  (10);  decimals  by  Latin 
prefixes:  deci  (1/10),  centi  (1/100),  milli  (1/1000).  Owing  to 
the  decimal  system,  few  measures  are  commonly  used.  Those 
used  are  shown  in  black-faced  type. 

Note.  Literature  on  the  Metric  System  is  published  by  The  Bureau  of 
Standards,  Washington,  D.  C. 


Linear  Measure 

1  myriameter 

=  10  kilometers 

=  10,000         meters. 

1  kilometer  (Km.) 

=  10  hectometers 

=  1,000 

1  hektometer 

=  10  decameters 

=      100 

1  dekameter 

=  10  meters 

10 

1  meter  (m.) 

=  10  decimeters 

=          1               " 

1  decimeter  (dm.) 

=  10  centimeters 

.1            " 

1  centimeter  (cm.) 

=  10  millimeters 

.01 

1  millimeter  (mm.) 

= 

.001        " 

182 


THE   METRIC   SYSTEM. 


183 


sq. 


Square  Measure. 
myriameter  =  100  sq.  kilometers       =  100,000,000 


sq.  meters. 


1  sq.  kilometer  (Km.')  =  100  sq.  hektometers 

=     1,000,000 

1  sq.  hektometer 

=  100  sq.  dekameters 

10,000 

1  sq.  dekameter 

=  100  sq.  meters 

100 

1  sq.  meter  (m.^) 

=  100  sq.  decimeters 

1 

1  sq.  decimeter  (dm.')    =  100  sq.  centimeters 

.01 

1  sq.  centimeter 

=  100  sq.  millimeters 

.0001       " 

1  sq.  millimeter 

Land  Measure. 

.000001   " 

1  hektare  (ha) 

=  100  ares                        =  10,000  sq.  meters. 

1  are 

=  100  centares                 = 

100  " 

1  centare 

Cubic  Measure. 

1  " 

1  cu.  myriameter 

=  1000  cu.  kilameters      =1,000,300,000,000  cu.  meters 

1  cu.  kilometer 

=  1000  cu.  hektometers  = 

1,000,000,000    "       " 

1  cu.  hektometer 

=  1000  cu.  dekameters    = 

1,000,000    "       " 

1  cu.  dekameter 

=  1000  cu.  meters 

1,000    "       " 

1  cu.  meter 

=  1000  cu.  decimeters     = 

1  "     " 

1  cu.  decimeter 

=  1000  cu.  centimeters    = 

.001               "       " 

1  cu.  centimeter 

=  1000  cu.  millimeters    = 

.0000001        "       " 

1  cu.  millimeter 

Table  of  Weight 

.000000001    "       " 

1  metric  ton  (t.)  (t 

onneau)     =10  quintals 

=  1,000,000        grams. 

1  quintal  (q.) 

=  10  myriagrams 

=    100,000 

1  myriagram 

=  10  kilograms 

=      10,000 

1  kilogram  (Kg.) 

=  10  hektograms 

1,000 

1  hektogram 

=  10  dekagrams 

100 

1  dekagram 

=  10  grams 

10 

1  gram  (g.) 

=  10  decigrams 

=         1 

1  decigram 

=  10  centigrams 

.1 

1  centigram  (eg.) 

=  10  miUigrams 

.01 

1  milligram  (mg.) 

= 

.001       " 

The  metric  ton 

or  tonneau=  weight  of  1  n 

I.  of  water;  the  kilogram  = 

weight  of  1  hter  of  water. 

Table  of  Capaciti; 

1  hektoliter  (HI.) 

=  10  dekaliters 

=  100      liters. 

1  dekaliter 

=  10  liters 

=   10 

1  Hter  (1.) 

=  10  deciliters 

=     1 

1  deciUter 

=  10  centiliters 

=       .1       " 

1  centiliter  (cl.) 

=  10  milliliters 

=       .01     " 

1  milliliter  (ml.) 

= 

.001  " 

184 


BUSINESS   ARITHMETIC. 


Table  of  Equivalents. 

Note.  Importers  and  customs  officials  have  to  perform  a  relatively 
slight  amount  of  reduction  to  and  from  the  metric  system.  Common 
equivalents  follow. 


Measure. 
Meter 

Liter 

Gram 
Kilometer 
Hectare 
Sq.  meter 
Cu.  meter 
Metric  ton 
Kilogram 
Mile 
Yard 
A. 

Sq.  yd. 
Cu.  yd. 


For  fairly  accurate  work.  For  approximation. 

39.37  in 3J  ft.  or  1.1  yd. 

f  1.0567  1.  qt.  ■) 

1    .9081d.  qt.J ^'^*' 

15.432    gr 15^  gr. 

.6214  mi 6  mi. 

2.47     A 2^  A. 

1.196    sq.  yd T.   1^  sq.  yd. 

1.308    cu.  yd. . .  ?f 1^  cu.  yd. 

1.1023  T 1.1  T. 

2.2046  1b 2.2  1b. 


1.6093  km. 

Quart  (liquid) 

.9436  1. 

.9144  m. 

Quart  (dry) 

1.101    1. 

40.47      a. 

Pound 

.4536  kg 

.8361  m2 

Ton 

.9072  t. 

.7645  m3 

ORAL    EXERCISE. 

1.  Read  as  meters: 

a.  7  km.,  5  dekameters  and  3  cm. 

b.  9  km.,  4  cm.,  8  mm. 

2.  Read  as  square  meters: 
a.  9  km2,  2  m^,  3  dm^. 
h.  5  hm2,  8  m^,  9  cm^. 

3.  Read  as  grams:  4  kg.,  8  dkg.,  7  eg. 

4.  Read  as  liters:  9  hi.,  7  1.,  4  cl.,  8  inl. 

5.  Use  decimal  values: 

100  m.  =  ?  in.  1000  m.  = 

1000  1.  =  ?  1.  qt.  200  t.    = 

2  km.  =  ?  mi.  10,000  m2  = 

6.  Use  approximate  equivalents: 
20  m.  =  ?  ft.  20 1. 
68.5 1.  =  ?  qt.  200  ha.     = 
80  km.  =  ?  mi.  600  m^     = 
40  kg.  =  ?  lb.  200  g.       = 

7.  Express  in  metric  system:  100  1.  qt.;  1000  lb. 
10  T.;  200  gr.;  100  cu.  yd.;  10  A.;  200  sq.  yd. 


?  cu.  yd. 
?  T. 
?  sq.  yd. 


=  ?  T. 
=  ?  A. 

=  ?  sq. 
=  ?  gr. 
100  mi, 


yd. 

;  1000  d.  qt.; 


THE   METRIC   SYSTEM.  185 

8.  A  "hundred  meter"  dash  is  equivalent  to  a  race  of  ?  yd. 

9.  A  man  who  orders  1000  m.  of  dress  goods  should  receive  ?  yd. 

EXERCISE. 

1.  If  a  merchant  desires   1200  yd.  of  French  silk,  he   must  order  ? 
meters. 

2.  A  consgt.  of  50  1.  olive  oil  will  fill  how  many  pint  bottles  ? 

3.  A  measurement  on  a  foreign  map  is  found  by  scale  to  be  equivalent 
to  87.56  km.     The  equivalent  distance  is  ?  miles. 

4.  An  importer  is  required  to  pay  a  tax  of  15c  per  gal.  on  120  1.  of  oil. 
The  tax  amounts  to  S?. 

5.  How  does  a  speed  of  71  km.  per  hour  for  a  German  express  train 
compare  with  a  record  of  71  miles  in  69  min.  made  by  a  train  in  this  country? 

6.  Granite  is  2.7  times  as  heavy  as  water.     Estimate  the  weight  of  a 
piece  measuring  3.1  m.  by  6  m.  by  4.55  m. 

7.  The   United  States  requires  foreign  letters  weighing  15  g.  or  less 
to  pay  5c  postage.     What  is  the  Umit  of  weight  in  our  standards? 

8.  What  is  the  mileage  rate  in  cents  for  first  class  passage  from  Rouen 
to  Paris,  136  m.,  if  the  fare  is  15.20  fr.  (1  fr.  =$.193). 

9.  A  man  who  weighs  172  lb.  weighs  ?  kg. 

10.  Determine  the  weight  of  an  aluminum  bar  1  m.  long  and  30  cm. 
cross-section  (specific  gravity,  2.71)? 

11.  A  tank  measuring  5'X4'X8'  has  a  capacity  of  ?  m*. 

12.  Determine  reasonable  dimensions  for  a  rectangular  tank  to  have 
a  capacity  of  500  1. 

13.  How  many  tons  of  coal  will  fill  an  order  for  5000  t.? 

14.  How  many  pieces  8.1  cm.  long,  may  be  cut  from  70  m.  of  wire? 

15.  A  400  meter  running  track  is  ?  yd.  long. 

Individual  Original  Work. 

1.  Prepare  a  brief  on  the  metric  system  contrasting  it  with  our  own 
system.     Illustrate  by  parallel  computations. 

2.  Construct  a  six-place  table  for  converting  Linear  Measure  to  Metric 
Measure. 

3.  Show  how  an  increasing  error  is  caused  in  the  construction  of  a 
working  table,  by  a  minor  error  at  the  start. 

4.  Compare  computation  by  table  and  by  process,  giving  illustrations 
to  show  the  value  of  tables. 


CHAPTER   XXVII. 
RATIO  AND   PROPORTION. 

INTRODUCTORY    EXERCISE 

1.  Find  the  difference  between  150  and  30. 

2.  30  is  what  part  of  150? 

3.  150  is  how  many  times  30? 

4.  $3000  worth  of  a  novelty  were  sold  in  one  year,  and  $12,000  worth 
the  next  year.  Find  the  increase  in  sales.  The  second  year's  sales  were 
how  many  times  those  of  the  first  year? 

It  is  evident  that  division  oflFers  one  effective  means  of 
comparing  numbers.  By  it  one  may  determine  how  many  times 
as  greaty  not  how  much  greater  one  number  is  than  another. 
The  numbers  so  compared  must  be  Hke  numbers,  and  conse- 
quently the  quotient  is  always  abstract.  This  quotient, 
whether  represented  or  computed,  is  termed  a  ratio. 

The  ratio  may  be  expressed  by  any  of  the  signs  of  division, 
or  by  the  colon.  The  numbers  involved  are  called  terms,  the 
first  term  being  the  antecedent  and  the  second  the  consequent. 

Illustration.  The  ratio  of  a  payment  of  $500  to  one  of  $2500  may  be 
expressed  (a)  $500/$2500,  (6)  $500 -^  $2500,  or  (c)  $500  :  $2500. 

The  value  of  the  ratio  is  determined  by  performing  the  repre- 
sented division.  In  the  above  illustration,  the  value  is  1/5. 
If  the  second  payment  had  been  referred  to  the  first,  the  ratio 
would  have  been  expressed  $2500  :  $500,  and  its  value  would 
have  been  5.  The  ratio  of  the  first  quantity  to  the  second  is 
termed  the  direct  ratio;  that  of  the  second  to  the  first  the 
inverse  ratio. 

ORAL   EXERCISE. 

Express  and  determine  the  ratios  of: 
1.    6  to  42.  2.    2  yd.  to  2  ft.  ' 

186 


RATIO  AND   PROPORTION.  187 

3.  5  qt.  to  6  gal.  8.  1'  to  1/4'. 

4.  2  sq.  ft.  to  1  sq.  yd.  9.  2  lb.  to  4  oz. 

5.  $720  to  $900.  10.  900  to  1500. 

6.  75c  to  $2.  11.  Measure  a  rod  by  a  yard. 

7.  1"  to  1'.  12.  Measure  a  bushel  by  three  pinta 

13.  Name  groups  of  two  numbers  having  the  ratios  12,  4,  1/2,  1^. 

14.  Add  1  to  the  antecedent  of  the  ratio  2:5.  What  is  the  effect  on 
the  value  of  the  ratio? 

15.  By  experiment,  find  the  effect  on  the  value  of  the  ratio,  from: 

(1)  Increasing  its  antecedent. 

(2)  Decreasing  its  antecedent. 

(3)  Increasing  its  consequent. 

(4)  Decreasing  its  consequent. 

(5)  Multiplying  or  dividing  the  antecedent. 

(6)  Multiplying  or  dividing  the  consequent. 

(7)  Multiplying  or  dividing  the  consequent  and  antecedent  by 

the  same  value. 

16.  Compare  the  above  cases  with  the  corresponding  cases  of  simple 
fractions. 

^.        Antecedent  (dividend)       -r^     .     ,         .... 

bince  -i^ ^  . ,.  . — ^  =  Ratio  (quotient),  it  is  evi- 

Consequent  (divisor) 

dent  that  the  antecedent  equals  the   product  of  ratio  value 

and  consequent,  and  that  the  consequent  equals  the  quotient 

of  the  antecedent  divided  by  the  ratio.     It  is  therefore  a  simple 

matter  to  find  a  single  missing  term. 

Illustration.  A  manufacturer  issues  one  design  of  oblong  calendars 
in  assorted  sizes,  but  with  the  dimensions  of  length  and  breadth  in  a  constant 
ratio  of  2|.  (a)  If  the  length  of  one  is  15"  what  is  the  width?  (6)  If 
the  width  of  one  is  8"  what  is  the  length? 

Solution  (a).    „„?^,    =  ratio  =  2|. 
Widtn 

Width  =Length-^2^  =  15"X2/5  =  6'^ 

Solution  (6).    Length = 2^  X the  width  =  2^X8"  =20". 

ORAL    EXERCISE. 

Determine  the  missing  values  in  each  of  these  ratios : 

1.    24  :4=  ?  2.     ?/18  =  3.  3.    36/?  =  2. 


188  BUSINESS   ARITHMETIC. 

4.  ?  :  4  =  8.  6.    3/4  :  2/3  =  ?  8.     1/4  yd.  :  ?  =  1/8. 

5.  1/2  : 1/4  =  ?  7.     ?  :  3  mi.  =  1/4.         9.    2h  ft.  :  U  ft.  =  ? 

Note.  If  two  variable  (changing)  quantities  or  dimensions  have  an 
unchanging  or  constant  ratio,  one  is  said  to  vary  as  the  other.  Thus  the 
ratio  of  the  circumference  of  a  circle  to  its  diameter  is  always  3.1416,  no 
matter  what  the  size  of  the  diameter.  Quantities  are  said  to  vary  inversely 
if  one  increases  as  the  other  decreases,  in  a  constant  ratio.  In  common 
speech,  we  refer  to  approximate  changes  in  the  same  way. 

10.  What  do  we  mean  by  saying  that  summer  travel  from  the  big  cities 
varies  with  the  heat? 

11.  What  do  we  mean  when  we  say  that  prices  vary  with  demand? 

12.  What  do  we  mean  when  we  say  that  the  cost  of  production  of  a 
certain  article  varies  inversely  as  the  quantity  produced? 

The  ratio  method  is  often  used  in  geometry  and  its  appli- 
cations, in  physics  and  in  the  solution  of  many  business 
principles.  It  is  really  a  method  of  fractional  analysis  and  is 
used,  without  the  name,  throughout  this  book. 

EXERCISE. 

1.  The  specific  gravity  of  copper  is  8.9.  What  is  the  weight  of  three 
cubic  feet?     See  page  134. 

2.  Water  is  composed  of  hydrogen  and  oxygen  in  the  ratio  of  2  : 1. 
How  many  parts  of  each  gas  in  15,000  parts  of  water? 

3.  If  a  3"  line  in  a  drawing  is  used  to  represent  a  real  length  of  4'  9", 
what  is  the  ratio  of  real  to  represented  length? 

4.  A  piece  of  ore  weighs  1.3  kg.,  while  the  weight  of  an  equal  volume  of 
water  is  .2  kg.     Determine  the  specific  gravity  of  the  ore. 

5.  If  the  circumference  of  a  wheel  is  to  be  16  ft.  what  must  be  its 
diameter? 

6.  One  of  two  cubes  is  3/4  of  the  height  of  the  other.  Determine  the 
ratio  of  surfaces  and  volumes. 

7.  Show  by  ratio,  the  effect  on  volume  of  reducing  all  dimensions  of  a 
bin  measuring  16'  X  8'  X  4',  by  one-third. 

The  Lever. 

If  a  bar  is  supported  on 


mri 


1 

J,      a  fulcrum,  F,  a  weight,  W, 

attached   at  the  point   Y 


RATIO  AND   PROPORTION.  189 

may  be  balanced  by  the  application  of  a  sufficient  pressure  at 
the  point  X.  If  the  weight  is  between  the  pressure  point  and 
the     fulcrum,    pressure 


must  be  applied  upward. 


.k. 


Otherwise,  it  is  applied     ^  "T  ~~J\ 

downward.     If  the  dis-  D3  ^ 

tances   x  and  y  are  ex- 
pressed in  units  of  measure,  and  W  and  P  in  units  of  weight, 
it  is  a  law  of  physics  that  the  ratio  of  the   product  of  the 
weight  by  its  arm  to  the  product  of  the  pressure  by  its  arm  is 
equal  to  unity,  when  the  lever  balances.     In  other  words. 
Weight  (W)  X  weight  arm  (y) 


Pressure  (P)  X  pressure  arm  (x) 


=  1. 


Illustration.     If  the  fulcrum  in  the  first  figure  is  4  ft.  from  Y  and  6  ft. 
from  X,  what  pressure  supports  a  weight  of  12  lb.? 
Solution. 

By  the  law  of  levers,  p  =  1. 

48  =  6P. 
S  =  P. 
.'.  the  pressure  must  be  8  lb. 

EXERCISE. 

1.  The  fulcrum  being  at  one  end  of  a  6  ft.  bar,  what  pressure  bal- 
ances a  weight  of  80  lb.,  1  ft.  6  in.  from  the  end? 

2.  A  pressure  of  20  lb.,  applied  9  ft.  from  the  fulcrum,  supports 
what  weight  2  ft.  beyond  the  fulcrum? 

3.  Illustrate  how  the  principle  of  the  lever  is  applied  in  the  use  of  a 
crowbar.     Write  and  solve  an  illustrative  example. 

4.  Find,  by  investigation,  six  common  applications  of  the  lever  prin- 
ciple, and  prepare  and  solve  simple  problems  illustrating  them. 

5.  Where  must  one  place  a  support  under  a  12  ft.  bar,  in  order  that  it 
may  balance  weights  of  16  and  40  lb.? 

6.  A  brake  chain  is  connected  with  a  brake  lever  1  ft.  below  the  sup- 
porting pin.  The  brake  handle  is  6  ft.  above  this  support.  If  the  brake 
lever  is  pulled  with  a  force  of  40  lb.,  what  pressure  does  the  brake  exert? 
What  pressure  per  square  inch,  if  the  brake  shoe  measured  8  in.  by  8  in.? 


190  BUSINESS  ARITHMETIC. 

Proportion. 

An  equality  of  ratios  is  termed  a  proportion.  Thus 
6  lb./21  lb.  =  $12/$42  is  a  proportion  because  the  abstract 
value  of  the  ratios  is  the  same.  An  equality  of  several 
ratios,  as  3/4  =  $9/$12  =  18  lb./24  lb.,  is  termed  a  continued 
proportion.  An  equality  of  products  of  ratios,  as  1/12  X  5/4  = 
5/6  X  1/8,  is  termed  a  compound  proportion.  Continued  and 
compound  proportion  are  seldom  used  in  practical  arithmetic. 

In  any  proportion,  as  1  :  2  =  |8  :  $16,  the  first  and  fourth 
terms  (1,  $16)  are  the  extremes,  and  the  second  and  third 
(2,  $8)  are  the  means.  The  proportion  is  read  "  1  is  to  2,  as 
$8  is  to  $16."  Since  the  two  ratios  are  equal  in  numerical 
value,  and  since  both  the  extremes  and  the  means  include  a 
numerator  of  one  ratio  and  a  denominator  of  the  other,  it  is 
evident  that  the  product  of  the  means  equals  the  product  of  the 
extremes.     Thus  1  X  $16  =  $16,  and  2  X  $8  =  $16. 

It  is  evident,  also,  that  any  term  of  a  proportion  may  be 

found  if  the  other  terms  are  known.     The  first  or  third  terms 

are  always  the  products  of  the  opposite  ratio  by  the  second  or 

fourth  term  respectively.     The  second  term  equals  the  product  of 

the  first  term  hy  the  opposite  inverse  ratio.     To  avoid  multiplying 

by  concrete  numbers,  each  term  of  either  ratio  may  be  made 

abstract. 

X       15 
Illustrations.     (1)  In  the  proportion  ^  =  j^  find  x. 

Solution. 

^  =  ^-^^  =  2.     Evidently  1/6  of  x  =  i|,  and  a;  =  6  X  ^f . 
40  40  46 

(2)  In  the  proportion =  -^  find  x. 

X  D 

Solution. 

6  24 

X  =  $480  X  24  =  $120.    Evidently  $480  =  a;  X  ^ , 

andx  =  $480  -^  ^  =  $480  X  ^^  =  $120. 
b  z4 

Note.  Since  the  product  of  the  means  equals  the  product  of  the  ex- 
tremes, it  is  evident,  also,  that  any  mean  may  be  found  by  dividing  the 


RATIO   AND   PROPORTION.  191 

product  of  the  extremes  by  the  other  mean;  and  any  extreme  may  be  de- 
termined by  dividing  the  product  of  the  means  by  the  other  extreme. 
Thus,  in  illustration  (1),  writing  the  proportion  in  the  form  a;  :  6  =  15  :  45, 
it  is  seen  that  one  extreme  is  missing  and  that  its  value  is  equal  to  6  X  15, 
the  product  of  the  means,  divided  by  45,  the  other  extreme.  This  principle 
affords  a  simple  method  of  solution. 

EXERCISE. 

Find  the  missing  terms: 

1.  a;  :  12  =  $72  :  $18. 

2.  5:8::  14:? 

3.  275  lb.  :  125  lb.  =  $11  :  $x. 

4.  3  sq.  ft.  :  16  sq.  ft.  :  :  48c  :  ?   c. 

5.  1/4  :l/2  ::?  :3/8. 

6.  a;:12  ::7yd.  :480  yd. 

Two  quantities  are  directly  proportional  if  one  varies  directly 
as  another.  They  are  inversely  proportional  if  one  varies 
inversely  as  the  other. 

Illustrations.  (1)  If  one  is  traveling  on  mileage,  the  cost  of  his 
railway  passage  varies  directly  as  the  distance,  or  is  proportional  to  distance. 

(2)  Temperature  being  constant,  the  volume  of  a  gas  varies  inversely 
as  the  pressure,  that  is,  grows  less  at  the  same  rate  that  the  pressure  in- 
creases.    Thus  doubling  the  pressure  halves  the  volume. 

The  common  applications  of  proportion  are  those  already 
mentioned  under  simple  ratio.  In  solutions  by  proportion,  it 
will  be  found  more  convenient  to  place  the  unknown  quantity 
in  the  first  term,  although  any  term  may  be  used.  Solutions 
may  be  checked  by  unitary  analysis,  and  by  fractional 
methods. 

Illustration.    A  contractor  is  to  receive  $4872  for  the  construction 
of  2140  ft.  of  concrete  pipe.     What  has  he  earned  after  he  has  completed 
535  ft.? 
Solution. 

Let  X  equal  the  sum  earned. 

X  :  $4872  =  the  ratio  of  amount  earned  to  contract  price. 
535  :  2140  =  the  ratio  of  quantity  done  to  total  quantity.     Since 
the  amount  earned  depends  on  the  work  done,  the  ratios  must  be 
equal. 


192  BUSINESS  ARITHMETIC. 

:.x  :$4872  =  535  :  2140. 

X  X  2140  =  $4872X535.     (Products  of  means  and  extremes.) 

^^  848^X535   =«218. 

Check  Solution. 

The  part  of  work  done  is  535/2140,  or  1/4. 

.-.  the  contractor  is  entitled  to  1/4  of  $4872,  or  $1218.      . 

EXERCISE.     . 

1.  The  weight  arm  of  a  lever  is  16  ft.;  the  pressure  arm  is  9  ft.;  the 
weight  is  4000  lb.     Find  the  pressure  or  power  to  balance  it. 

2.  A  farmer  found  that  he  required  7  lb.  of  a  certain  seed  for  a  1/4 
acre  lot.     What  proportional  amount  of  seed  is  required  for  a  3^  acre  lot? 

Note,  In  electricity,  the  resistance  of  a  wire  to  the  flow  of  a  current 
varies  directly  as  its  length,  or  inversely  as  the  square  of  its  cross  section. 

3.  Determine  the  resistance  of  If  mi.  of  a  certain  copper  wire,  if  the 
resistance  of  100  yd.  is  5  ohms.  (The  ohm  is  an  electricity  unit  of  re- 
sistance.) 

4.  If  the  resistance  of  iron  is  seven  times  that  of  copper,  find  the  re- 
sistance in  1000  ft.  of  iron  wire,  if  20  yd.  of  copper  wire  of  same  diameter 
has  a  resistance  of  2.6  ohms. 

5.  If  a  metal  sphere  of  8  in.  diameter  weighs  214  lb.,  what  is  the  weight 
of  a  sphere  of  the  same  metal  having  a  diameter  of  12  in.? 

6.  It  is  a  principle  of  mechanics  that  the  power  acting  parallel  to  an 
inclined  plane  will  support  a  weight  proportional  to  the  ratio  of  the  length 
of  the  incline  to  the  distance  the  weight  is  raised.  If  the  height  is  5/6 
of  the  length,  what  power  will  support  a  weight  of  840  lb.?  If  the  height 
is  1/4  of  the  length? 

Note.    For  the  principle  of  partitive  proportion,  see  page  387. 


CHAPTER  XXVIII. 

GRAPHIC  ARITHMETIC. 

A  simple  drawing  often  does  .away  with  the  necessity  for  a 
compUcated  numerical  statement.  A  line  diagram,  drawn  to 
scale,  will  often  convey  its  meaning  instantaneously.  A  num- 
ber statement,  on  the  other  hand,  may  convey  little  meaning, 
or  require  time  for  interpretation,  or  be  less  likely  to  attract 
attention.  Diagrams  or  graphs,  therefore,  have  become  a 
common  form  of  number  expression,  or  language,  especially 
in  the  presentation  of  statistical  matter.     Thus  the  accom- 


Production  of  Barley:  1900 

MILLIONS   OF   BUSHELS 
8  12  16 


20 


24 


CALIFORNIA 

MINNESOTA 

WISCONSIN 

IOWA 

8. DAKOTA 

N.DAKOTA 

WASHINGTON 

NEW  YORK 

NEBRASKA 

OREGON 

KANSAS 

MICHIGAN 

OHIO 

IDAHO 

MONTANA 

ILLINOIS 

COLORADO 

ARIZONA 

VERMONT 

OKLAHOMA 

INDIANA 

MAINE 

UTAH 

NEVADA 

PENNSYLVANIA 

TEXAS 


14 


193 


194 


BUSINESS   ARITHMETIC. 


panying  graph  from  a  Census  Report  gives  the  average  reader 
a  far  clearer  impression  of  the  relative  production  of  barley, 
in  different  states,  than  would  a  column  of  figures  stating  the 
production  in  bushels.  Moreover,  if  desired,  the  graph  may 
be  read  to  approximate  numerical  value  by  means  of  its  scale. 
General  diagrams,  such  as  building  plans,  plots  of  land, 
maps,  etc.,  commonly  are  used  for  reference.  It  is  far  easier 
and  surer  to  say  "Find  the  area  of  the  floor  shown  in  the 
accompanying  diagram"  than,  without  a  diagram,  to  describe 
the  shape  and  dimensions  in  such  a  way  as  to  enable  one  to 
compute  its  area. 


Experiment.    Try  to  describe  the  plan  here  shown,  in  words  so  that 
one  could  compute  its  area  accurately 

I.    Scales  and  Plotting. 
Ability  to  read  and  construct  explanatory  drawings  and 
statistical  graphs  depends  on  a  thorough  knowledge  of  the 
arithmetic  of  scales,  and  of  the  fundamentals  of  plotting. 


GRAPHIC   ARITHMETIC.  195 

A  linear  scale  is  either  the  ratio  of  the  plotted  length  to  the 
true  length  or  distance,  or  the  length  that  is  to  represent  on 
paper  a  fixed  quantity  of  any  denomination.  If  every  line  in 
a  drawing  is  one-fourth  of  the  corresponding  true  length 
of  the  object  represented,  the  diagram  is  said  to  be  drawn  on 
"  1/4  scale. "  If  a  one-inch  line  is  used  to  represent  a  quantity 
of  10,000  bushels,  the  scale  is  said  to  be  *'  1  inch  =  10,000 
bushels." 

Linear  scales  are  represented,  commonly,  in  one  of  three 
ways: 

(1)  As  a  simple  fraction.     Illustrations:  1/4,  3/8,  5/4. 

(2)  By  corresponding  dimensions.  Illustration:  V  =  T. 
This  means  that  one  inch  on  paper  represents  one  foot  of  the 
object. 

(3)  By  diagram.     Illustration: 


10  5  0  10  20  30  40 

I  I  r  I  I  I  I  I  I  I  I \ I I I 

FEET 


From  zero  to  the  right  the  scale  shows  diagram  lengths 
corresponding  to  true  lengths  of  10  ft.,  20  ft.,  30  ft.,  etc.  A 
true  distance  of  22  ft.  is  represented  by  a  paper  distance  from 
"20"  (on  the  right  of  zero)  to  the  second  sub-division  on  the 
left.  (Why?)  Notice  that  the  paper  scale  reads  to  true 
lengths.     In  diagram  scales  decimal  sub-divisions  are  common. 

Linear  scales  of  distance  may  be  expressed  in  any  of  the 
three  forms.  Thus  a  scale  of  1/3  may  be  written  V  =  S", 
or  1  ft.  =  3  ft.;  or . 

3    INCHES 

QUESTIONS    FOR    DISCUSSION. 

1.  What  is  the  advantage  m  having  the  diagram  scale  read  to  true 
lengths? 

2.  Why  is  a  linear  scale  used  on  a  drawing  that  represents  an  area  ? 


196  BUSINESS  ARITHMETIC. 

3.  How  does  the  "line"  scale,  if  carefully  constructed,  save  compu- 
tation, in  making  or  reading  drawings? 

4.  Name  a  common  scale  used  by  architects. 

5.  How  does  the  graduated  edge  of  a  ruler  correspond  to  a  scale? 

EXERCISE. 

(Answer  orally,  if  possible.) 
Express  each  of  these  scales  in  two  other  forms : 

11.     1"  =  100'.  12.     2'  =  U". 

13.  Which  of  the  above  scales  could  be  used  to  make  a  diagram  larger 
than  the  original? 

14.  Express  the  following  scales  in  numerical  form.     (Use  ruler  to 
find  ratio.) 

(a)  Q>) 

t      5  FT.-    J  2p^     ,,,.,,,.   0^ . 2,0' 


9                             1 

(c) 

? 

f 

MILES 

id) 

-J 

1  // 
75- 


15.    Which  of  these  scales  will  make  the  largest  diagram: 
i--  1"  =  4'-  9. 


16'  '  FEET 

16.  An  engineer  uses  a  ruler  divided  into  units,  tenths  and  hundredths 
of  feet.  If  he  uses  the  smallest  subdivision  to  represent  two  feet,  find 
his  scale.     Express  it  in  two  ways. 

17.  If  the  smallest  subdivision  on  your  ruler  is  used  to  represent  one 
foot,  what  is  the  scale? 

18.  An  architect  uses  a  ruler  having  inches  divided  into  sixteenths. 
If  he  uses  one  of  these  subdivisions  to  represent  a  foot,  what  is  his  scale? 


GRAPHIC   ARITHMETIC.  197 

19.    Two  towns,  known  to  be  twelve  miles  apart,  are  represented  on  the 
map  at  an  interval  of  three  inches.     What  is  the  map  scale? 

EXERCISE. 

1.  Construct  the  scale  1/20  in  diagram  form. 

2.  Determine  a  fractional  scale  for  this    (ji_ i^o 

line  scale.  rods  - 

3.  Construct  a  line  scale  for  the  scale  V  =  1200  bbl. 

4.  Construct  a  line  scale  for  the  scale  1"  =  2500  lb.,  such  that  each 
division  shall  represent  1000  lb. 

5.  Find  and  illustrate  four  methods  of  representing  line  scales. 

Reduction  to  Plotted  Lengths  or  Distances.  Since  a  true 
linear  scale  shows,  in  fractional  form,  the  part  that  the  scale 
length  is  of  the  length  to  be  plotted,  computations  in  scale  reduc- 
tion are  simple  fractional  reductions.  A  scale  of  one-sixth 
means  that  the  plotted  lengths  are  one-sixth  of  the  real  lengths. 
Real  lengths,  in  this  case,  are  reduced  to  plotted  lengths  by 
dividing  by  six.  When  scales  are  expressed  in  other  forms, 
they  may  be  used  directly,  or  in  difficult 

n 5       cases,  reduced  to  fractional  form.     In  the 

f'^ET  scale  shown  five  feet  of  true  length  are  repre- 

sented by  the  given  line.  Twenty-five  feet 
would  be  represented,  evidently,  by  a  line  five  times  as  long. 
A  scale  of  V  =  4'  reduces  an  8'  true  dimension  to  a  plotted 
dimension  of  2".  (Why?)  On  a  scale  of  1"  =  $600,  the 
amount  of  $1200  is  represented  by  a  2"  line.     (Why?) 

EXERCISE. 
(Answer  orally  if  possible.) 

Find  the  scale  lengths  for  true  dimensions  of* 

1.  4'  6",  on  a  scale  of  1/2. 

2.  3'  9",  on  a  scale  of  I"  =  1'. 

3.  100  yd.,  on  a  scale  of  1/600. 

4.  35  ft.  on  a  scale  of  1"  =  10  ft. 


198  BUSINESS   ARITHMETIC. 

5.  $2000,  on  a  scale  of  1"  =  $400. 

What  length  lines  should  be  used  in  graphs  to  represent  these  quantities: 

6.  4500  bu.  on  a  scale  of  1"  =  250  bu. 

7.  4  tons,  on  a  scale  of  1/2"  =  500  lb. 

8.  1600  cans,  on  a  scale  of  1/4"  =  200  cans. 

9.  2,400,000  bales,  on  a  scale  of  1"  =  60,000  bales. 

EXERCISE. 

Find  the  missing  values: 
No.  True  dimension.  Scale.  Scale  Dimension. 

1.  4' 6"  ' '  ? 

X.  t.  u  2  FEET  * 

2.  2' 3"  1/12  ? 

3.  14'  9"  1/4"  =  1'  ? 

4.  270'  0"  1"  =  50'.  ? 

5.  4  mi.  80  rd.  '  TmilI '  ? 

6.  8"  5/3  ? 

7.  2 1/2"  1"  =  1/3"  ? 

8.  Which  of  the  above  scales  will  make  enlargements? 


PRACTICE    EXERCISE. 

1.  List  twenty  true  dimensions  taken  in  the  class  room.  In  a  parallel 
column,  write  the  corresponding  scale  lengths.  Use  the  smallest  sub- 
division on  your  ruler  to  represent  one  inch. 

2.  Make  a  freehand  sketch  of  the  outline  of  the  class  room  floor.  Take 
measurements  and  determine  plotted  lengths  on  a  scale  of  1/16. 

Dimensioning  is  the  process  of  noting  on  a  drawing  the 
true  dimensions  of  the  parts  represented  (page  194).  The 
symbols  (')  and  {")  are  commonly  used  in  place  of  the  abbrevi- 
ations (ft.)  and  (in.).  If  the  true  length  contains  no  fraction 
of  a  foot,  zero  inches  is  written.  For  example,  5  ft.  is  written 
5'  0".  Light  dimension  lines  are  drawn  parallel  to  the  lines 
of  the  drawing  to  which  the  dimensions  refer,  the  dimension 
figures  being  entered  in  a  break  in  the  center.     Dimension 


GRAPHIC  ARITHMETIC.  199 

lines  are  limited  by  arrow  points  that  just  touch  short 
perpendiculars  erected  at  the  extremes  of  the  line  to  be 
dimensioned.  Dimensions  are  generally  written  up  and  to  the 
right,  in  order  that  all  may  be  read  from  one  position.  A  circle 
or  arc  is  dimensioned  by  locating  its  center  and. giving  its 
radius.  "Over  all"  or  "total"  dimensions  are  written  as  a 
check  on  a  series  of  intermediate  dimensions. 

EXERCISE. 

1.  Point  out  the  different  principles  of  dimensioning  illustrated  on 
page  194. 

2.  What  is  the  real  value  of  an  "over  all"  dimension? 

3.  What  is  the  advantage  in  writing  dimensions  outside,  rather  than 
inside  a  small  drawing? 

4.  Collect  from  newspapers,  periodicals,  or  other  sources,  six  specimens 
of  dimensioned  drawings.  Make  a  list  of  any  unusual  details  of  dimen- 
sioning. 

Plotting,  as  related  to  drawings,  is  the  process  of  locating 
the  main  points  of  a  proposed  diagram  preliminary  to  drawing 
in  the  details.  With  the  actual  mechanical  construction  of 
drawings  we  have  little  to  do,  but  certain  mathematical 
preliminaries  should  be  noticed.  Before  plotting,  it  is  neces- 
sary to  choose  a  paper  large  enough  to  contain  the  drawings 
of  the  object  at  the  scale  required.  The  size  of  paper  and 
the  dimensions  of  the  object  to  be  represented  will  necessarily 
affect  the  scale  to  be  chosen. 

EXERCISE. 

1.  The  greatest  dimensions  of  a  tract  of  land  are  1800  ft.  by  1450  ft. 
If  plotted  on  a  scale  of  1/450,  how  large  a  sheet  is  required,  allowing  for  a 
two- inch  margin  all  around? 

2.  What  is  the  largest  scale  that  could  be  used  in  reproducing,  on 
this  page,  the  map  of  a  piece  of  land  measuring  4  miles  by  3  miles  40  rods? 

3.  What  is  the  largest  scale  on  which  the  land  diagram  on  page  149 
could  have  been  drawn? 

4.  What  is  the  largest  scale  on  which  a  diagram  of  your  class  room 
could  be  drawn,  on  a  paper  measuring  17  in.  by  24  in.? 


200  BUSINESS  ARITHMETIC. 

5.  A  rectangle  measures  16  ft.  by  5  ft.  2  in.  How  large  a  sheet  (A 
paper  is  required  for  a  drawing  of  it,  on  a  scale  of  1/4"  =  1',  if  a  1^" 
margin  is  to  be  left  between  the  drawing  and  the  edges  of  the  paper? 

Coordinates.  While  the  general  principles  of  scientific 
projection  and  plotting,  as  used  in  projection  drawing,  have  no 

place  in  this  discussion,  the 
Y  system  of  plotting  by  coor- 

dinates has  a  direct  use  in 
+3  Pa  the  graphic  phases   of  ap- 

1+5   p  plied  arithmetic.     If  we  as- 

ffs  1+4  sume  two  fixed  lines,  Y  and 

^^\!     j  T  X,    at    right    angles,    any 


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point,  P,  may  be  exactly 
located  if  its  perpendicular 
distances  from  X  and  Y  are 
known,  together  with  the 
particular  angle,  I,  II,  III, 
^  or  IV,  in  which  it  is  situ- 

ated. If  P  is  in  angle  (quad- 
rant) I,  4  units  from  X  and  5  from  Y,  it  is  located  by 
measuring  on  X  from  Y  5  units;  then  up  from  X,  parallel  to  X, 
4  units.  The  coordinates,  or  locating  distances,  are  always 
taken  parallel  to  the  axes  of  coordinates  {X  and  F),  so  that  it 
makes  no  fundamental  difference  whether  the  axes  are  at  right 
angles  or  at  a  slant.  In  writing  (not  plotting),  the  position 
of  a  point  with  respect  to  the  distance  parallel  to  X  is  stated 
first,  then  the  distance  parallel  to  Y.  Thus  the  point  P  is  the 
point  (5,  4);  the  point  Pi  is  (2,  1);  the  point  P2  is  (?,  ?);  the 
point  Pe  is  (0,  3).  If  the  point  is  located  in  other  quadrants, 
its  position  is  located  in  the  same  way,  except  that  the  dis- 
tances parallel  to  X  and  to  the  left  of  Y;  or  parallel  to  Y  and 
below  X  are  marked  minus.  Thus  P3  is  the  point  (—2,  5); 
P4  is  the  point  (—3,  —5),  and  Ps  the  point  (3,  —5).  In  the 
simpler  applications,  with  which  we  have  to  deal,  it  is  possible 


GRAPHIC  ARITHMETIC. 


201 


so  to  locate  the  axes  as  to  have  all  our  points  in  the  same  angle, 
and  thus  disregard  the  plus  and  minus  signs. 


EXERCISE. 

Draw  the  axes  and  locate  these  points: 

1. 

2  in.  from  X  and  3  in.  from  Y,  first  quadrant. 

2. 

4  in.  from  Y  and  3  in.  from  X,  first  quadrant. 

3. 

On  X  2  m.  from  Y. 

4. 

(3",  3").                   7.     (2",  -4").     ■                10. 

(0',  00. 

5. 

(4",  2").                   8.     (0",  -U").                    11. 

(3/4-,   -2"). 

6. 

(2^",  5'0.                9.     (5",      0").                     12. 

(1",  2h"). 

13. 

(8",  5"),  on  1/2  scale. 

14. 

(2  mi.,  7  mi.)  on  a  scale  of =  2  miles. 

If  one  locates  the  corners  of  a  field  with  respect  to  axes,  he 
may  plot  the  points  and,  by  connecting  them,  obtain  the  out- 
line. In  taking  measurements 
of  an  irregular  room,  one  may 
find  the  distances  of  each  cor- 
ner from  any  side  or  sides, 
taken  as  axes,  and  then  pre- 
pare a  plot.  Points  on  a  curve 
may  be  taken  at  such  close 
intervals  as,  after  plotting,  to 
outline  the  curve  on  paper. 
As  a  help  in  plotting,  cross- 
section  paper,  similar  to  the  il- 
lustration, may  be  used,  any  two  lines  at  right  angles  being 
taken  as  axes.  The  sides  of  the  sub-squares  may  then  be  taken 
as  units  of  scale,  and  diagrams  plotted  speedily.  The  large 
squares  are  usually  divided  decimally. 


"~ 

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_^ 

^ 

EXERCISE. 

1.     Plot  the  points  (4,  3),  (7,  3)  and  (6,  6). 
the  resulting  figure? 


Connect  them.    What  is 


202  BUSINESS  ARITHMETIC. 

2.  Plot  to  scale  the  points  (2,  2),  (3,  4),  (6,  2)  and  (7,  4).  What  is 
the  resulting  figure? 

3.  Make  a  diagram  on  a  scale  of  1"  =  6',  of  a  room  whose  corners, 
taken  in  order,  are  (0',  0'),  (0',  11'),  (8',  11'),  (8',  13'),  (13',  13')  and  (13',  0'). 
Which  sides  are  taken  as  axes. 

4.  Make  a  diagram  of  your  class  room  by  the  same  process. 

5.  Plot  and  find  the  area  of  a  field  whose  comers,  taken  in  order,  are 
the  points  (50  rd.,  70  rd.),  (200  rd.,  70  rd.),  (250  rd.,  200  rd.)  and  (50  rd., 
200  rd.). 

6.  Plot  one  side  of  a  curve  of  a  driveway  from  the  points  (0',  10') 

(1,  8),  (2i  6),  (5,  4),  (6,  3),  (8,  2),  (10,  1). 

Reduction  of  Plotted   Lengths   to   Real    Lengths.      Few 

drawings  or  maps,  except  draftsmen's  construction  drawings, 
are  so  fully  dimensioned  as  to  supply  all  dimensions  desired. 
It  is  often  necessary  to  find  the  real  length  corresponding  to 
a  plotted  length.  If  a  plotted  length  is  drawn  on  a  one-fifth 
scale,  the  true  length  is  naturally  ^ve  times  that  plotted. 
If  the  drawing  scale  is  V  =  4',  a  2''  line  on  paper  represents 
8'  of  real  length.  (Why?)  If  a  scale  is  2  feet,  a  line 
six  and  one-half  times  as  long  represents  a  true  length  of 
13  feet.     (Why?) 

EXERCISE. 
(Oral  and  written.) 
No.        Length  on  Drawing.  Scale.  True  Lengths 

1.  3i"  1"  =  1000'.  ? 

2.  '         5.3"  1/1200  •? 

3.  2.7"  1/63360  ? 

4.  1',  3.4"  2  INCHES'  ^ 

5.  9.2"  1"  =  2  mi.  ? 

6.  1',  3^'  1/16  ? 

7.  11.45"  1/400  ? 

8.  2^'  1"  =  $5250  ? 

9.  4i"  1/2"  =  10,000  bu.  ? 
10.      1"  =  10'.  ? 


GRAPHIC   ARITHMETIC. 


203 


11.  Determine  the  scale  of  the  figure  on  p.  155.  Find  the  true  distance 
diagonally  across  the  room;  and  the  distances  of  any  point  C  from  each 
comer,  and  perpendicularly,  from  three  sides. 


STATE         LINE 


Avondale 


Larsliall 


12.  This  map  is  drawn  at  a  scale  of  1"  =  ?  mi.,  or  what  fractional  scale? 
Which  towns  are  in  a  radius  of  eight  miles  of  the  center  of  Marshall? 
By  airline,  Mason ville  is  ?  miles  from  Avondale;  Norwood  is  ?  miles  from 
the  nearest  point  on  a  railway;  Mason  ville  is  ?  miles  from  the  state  line, 
and  Norwood  is  ?  miles  from  the  nearest  railway  junction.  An  airline 
electric  road  from  Ashland  to  Marshall  would  be  ?  miles  long,  approxi- 
mately ?  miles  shorter  than  the  nearest  railway  connection. 

Are  distances  computed  by  scale  from  this  map  absolutely  accurate? 
What  factors  tend  to  affect  the  results? 


204 


BUSINESS   ARITHMETIC. 


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GRAPHIC  ARITHMETIC. 


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BUSINESS   ARITHMETIC. 


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AVE. 


Scale? 


13.     The  owners  of  the  plot  here  shown  desire  to  buy  additional  land, 
fronting  60'  on  13th  Street  and  80'  adjoining  on  Adams  Ave.,  and  running 

back  in  each  case  so  as  to  make 
the  entire  lot  a  rectangle  measur- 
ing V  by  ?'.  Plot  the  figure  and 
show  the  additions.  Locate  on  the 
plot  a  building  with  a  frontage  par- 
allel to  Adams  Ave.,  of  80',  set 
back  20'  from  the  street  line,  and 
running  back,  parallel  to  13th 
street,  to  a  depth  of  60'.  The 
building  is  15'  from  the  13th  Street 
line.  The  area  of  the  original  plot 
is  ?  sq.  yd.;  of  the  addition,  ?  sq. 
yd. ;  and  of  the  improvement  ?  sq. 
ft. 
14.  Sixth  Street,  north,  meets  Ave.  A  at  right  angles.  On  a  scale  of 
1/1200,  make  a  plot  of  the  land  on  the  northwest  corner,  from  this  de- 
scription: From  the  corner,  the  boundary  runs  north  180';  thence 
west,  40' ;  thence  north,  60' ;  thence 
west,  80';  thence  diagonally  to  a 
point  on  Ave.  A  100'  from  corner. 
The  area  is  —  sq.  ft.,  and  the  valu- 
ation, at  $1.95  per  sq.  ft.  is  $ . 

II.  Graphs. 
Reference  has  been  made 
to  the  graph  as  a  means  of 
conveying  general  impres- 
sions of  relative  value.  It 
deserves,  however,  more  ex- 
tended notice,  now  that  the 
principles  of  scales  are  un- 
derstood. 


t 


What  We  Have        What  We  Grow        What  We  Use 


OUR  TIMBER  SUPPLY  AND  ITS  DEPLETION. 
The  large  tree  represents  the  amount  of  timber  that  we  have,  the  second 
tree  the  relative  amount  of  the  annual  growth,  and  the  third  tree  the  relative 
amount  of  annual  use. 

From  The  World^s  Work. 


GRAPHIC   ARITHMETIC.  207 

The  graph  may  be  used  in  the  form  of:  (1)  lines  drawn  to 
scale  in  lengths  proportional  to  the  quantities  of  bushels,  dol- 
lars, acres,  etc.,  which  they  represent;  (2)  proportional  areas 
of  any  shape;  (3)  lines  based  on  the  principle  of  coordinate;  or 
(4)  figures,  varying  in  one  or  more  dimensions.  xAn  illustration 
of  (4)  is  shown  on  page  206. 

The  Parallel  Line  Graph  is  often  drawn  superposed  on  a 
reading  scale  that  enables  one  to  read  approximate  numerical 
values  at  sight.  In  the  graph  of  the  production  of  corn  (p.  204) 
in  the  United  States,  based  on  Government  figures,  each  scale 
division  represents  one  hundred  million  bushels.  Thus  the  crop 
for  1910  was  approximately  3,125,000,000  bushels  and  for 
1850  it  was  just  under  600,000,000  bushels.  Approximate 
values  are  all  the  general  public  desires,  hence  for  large  values 
round  numbers  are  sufficient.  Such  graphs  also  show  at  a 
glance  the  general  increase  of  crop  from  period  to  period  and 
any  period  of  unusual  crop  or  crop  shortage. 

By  using  rectangles  of  a  uniform  height  so  that  areas  vary 
with  widths,  the  principle  of  the  line  graph  is  maintained, 
but  changes  in  the  filling  of  the  different  sections  are  made  to 
add  details  of  information.  Thus  in  the  figure  on  p.  205 
each  entire  space  represents  the  full  arid  area  of  a  state,  but, 
by  a  change  of  shading,  five  facts  concerning  this  area  are 
made  clear. 

Area  graphs  are  of  endless  variety  but  they  usually  conform 
to  geometrical  figures.  In  the  circular  graphs  (p.  208),  areas 
vary  with  represented  values.  Both  were  constructed  to  scale, 
although  none  is  stated. 

The  temperature  and  precipitation  chart  shown  on  p.  209 
illustrate  a  coordinate  graph.  The  subdivisions  of  the  scale 
background  vertically  read  represent  (1)  temperatures  in 
degrees  above  or  below  the  normal  for  the  periods  shown,  and 
(2)  the  precipitation  in  tenths  of  inches.     The  solid  lines  show 


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GRAPHIC   ARITHMETIC. 


209 


the  variation  in  average  temperature  from  week  to  week, 
from  the  normal  as  shown  by  the  horizontal  line.  The 
dotted  line  shows  the  amount  and  variation  (departure)  in 
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||               TEMPERATURE,  IN  DEGREES,  PRECIPITATION,  IN  TENTHS  OF  IN DHE3, ^_     j| 

Temperature  (degrees  Fahrenheit)  and  precipitation  (inches)  depart- 
ures for  the  season  of  1905  from  the  normal  of  many  years  for  the  Middle 
and  South  Atlantic  States  and  Gulf  States. 

From  the  Year  Book  of  the  U.  S.  Department  of  Agriculture  for  the 
year,  1905. 

EXERCISE. 

1.  Read  at  sight  the  figure  for  the  production  of  com,  page  206, 

2.  Reduce  to  a  number  table  the  graph  shown  on  page  206. 

3.  Read  the  following  graph  at  sight. 

PRODUCTION  OF   COTTON:    1850-1900. 


10 


( 

1900 
1890 
1880 
1870 
1860 
1850 

3 

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MILLIONS   OF 
3              4 

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15 


2m  BUSINESS  ARITHMETIC. 

4,  The  corresponding  1910  figure  for  the  above  cotton  graph  may  be 
taken  as  10,386,000  bales.  Determine  the  length  of  the  proper  line  on  the 
graph. 

5;  A  man,  receiving  a  wage  of  $25.00  per  week,  estimates  his  expenses 
as  follows:  Rent,  $25  per  month;  food,  $360  per  year;  fuel  and  Ught  $80 
per  .year;  clothing  $100  per  year;  amusements,  $2  per  month;  charity  $2 
per  month;  miscellaneous  expense,  $6  per  month — the  balance  remaining 
representing  a  savings  and  emergency  fund.  Draw  an  outline  to  represent 
his  $20  bill  or  weekly  wage,  and  show  graphically  how  it  is  apportioned  to 
meet  his  expenses. 

6.  The  following  have  been  estimated  to  be  the  approximate  expendi- 
tures of  anthracite  coal  in  New  York  City  in  a  year: 

Anthracite.  Tons. 

Domestic,  private  houses  and  small  stores 2,500,000 

Flats  and  apartment  houses .' 2,850,000 

Hotels,  clubs  and  theaters 1,250,000 

Gas  and  electric  plants 1,300,000 

Elevated  and  surface  roads 650,000 

Harbor  shipping 400,000 

Department  stores  and  office  buildings 650,000 

Municipal  departments 400,000 

Total 10,000,000 

Represent  these  figures  graphically  in  some  other  way  than  by  a  line  graph. 

7.  By  means  of  properly  proportioned  outline,  to  suggest  "  skyscrapers," 
represent  the  relative  amount  of  new  building  operations  in  different  cities, 
in  1911  based  on  the  following  figures:  Baltimore,  $9,325,000;  Buffalo, 
$10,365,000,  Chicago,  $105,270,000;  Cincinnati,  $13,485,000;  Cleveland, 
$17,000,000;  Detroit,  $19,000,000;  New  York,  $173,500,000;  Pittsburgh, 
$11,700,000. 

8.  Represent  in  some  other  graphic  form  the  graph  on  page  211. 

9.  Using  round  numbers,  to  the  nearest  "ten  thousand,"  prepare  a 
graph  to  illustrate  these  facts  concerning  immigration: 

Immigration  from  1830  to  1906. 

1831-1840 599,125 

1841-1850 1,713,251 

1851-1860 2,598,214 

■  1861-1870 2,314,824 

1871-1880 2,812,191 

1881-1890 5,246,613 

1891-1900.  ;  .  . 3,844,420 

1901-1910 .8,795,386 


GRAPHIC  ARITHMETIC. 


211 


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212  BUSINESS  ARITHMETIC. 

10.  Design  a  graph  to  illustrate  this  statement:  "The  llama  will  cany 
from  50  to  200  pounds;  a  man  from  75  to  150  pounds;  the  donkey,  100  to 
200  pounds;  an  ox,  150  to  200  pounds;  a  horse  from  200  to  250  pounds; 
the  camel  from  350  to  500  pounds;  the  elephant,  from  1800  to  2500  pounds." 

11.  Construct,  on  a  coordinate  or  cross-section  plan,  a  graph  to  rep- 
resent the  recapitulation  of  daily  sales  by  departments,  shown  on  page  13. 
Design  a  form  that  may  be  extended  day  by  day. 

Individual  Original  Work. 
Prepare  a  set  of  statistics  of  some  class  of  the  high  school,  compiled  from 
office  records  by  direction  of  Principal.     Show  same  facts  graphically. 


CHAPTER   XXIX. 
PERCENTAGE. 

In  the  expression  of  numerical  relations,  100  is  a  common 
basis  of  comparison.  The  process  of  percentage,  or  of  com- 
puting by  hundreds,  is  one  of  the  most  important  of  all  the 
working  processes  of  arithmetic  and  has  most  varied  applica- 
tions. 

INTRODUCTORY    EXERCISE. 

1.  Find  1/100  of  200,  2472,  36,  592  bu.,  h  .006. 

2.  Find  3/100  of  400,  70  lb.,  9000  tons. 

3.  Find  (a)  4/100  of  720;  (b)  25/100  of  800  sheep. 

Per  cent,  is  a  standard  term  for  hundredths,  derived  from  the 
Latin  per  centum,  meaning  "  by  the  hundred."  The  term  is 
represented  by  the  symbol  "%." 

Illustration.    4  per  cent,  of  a  number  =  4%  of  it  =  4/100  of  it  =  .04 

of  it. 

ORAL    EXERCISE. 

1.  What  per  cent,  of  a  quantity  is  one-hundredth  of  it?     3/100  of  it? 
246  hundredths  of  it?     .03  of  it?     .92  of  it?     3/4  of  it? 

2.  Express  as  rates  per  cent. :  36/100,  270/100,  .40,  .02,  1.25,  3.205,  .6. 
Note.     The  rate  per  cent,  is  the  number  of  hundredths  taken. 

3.  Express  as  hundredths:   8%,    12%,    120%,    ^%,   331%,   600%, 
121%. 

4.  i  =  ?  hundredths.     ^  of  a  quantity  =  ?  %  of  it. 

5.  Express  as  per  cents.:  1/3,  3/4,  1/4,  5/8,  3/20,  1/40,  3/5. 

6.  20%  of  a  quantity  equal  how  many  himdredths  of  it?    What 
fractional  part  of  it? 

7.  Name  simple  fractions  corresponding  to  these  per  cents. :  50%,  5%, 
25%,  40%,  36%,  1331%,  100%,  106%,  96%.. 

8.  Express  as  decimals:  1/4,  1/3,  5/4,  3/25,  4.3%,  125%,  20%,  12i%, 
92%. 

9.  Express  as  condmon  fractions  .24,  .4,  3.2,  .0125,  90%,  120%,  43%. 
10. .  Express  each  of  these  rates  per  cent,  in  at  least  two  other  forms: 

.26,  .8,  1  1/4%,  .004,  9/5,  3/4,  1/2%,  .50,  48%,  75%. 

213 


214  BUSINESS  ARITHMETIC. 

11.  Express  these  statements  in  other  arithmetical  forms: 

(a)  I  pay  33^%  of  a  debt. 

(6)  The  use  of  wood  pulp  paper  in  this  country  has  increased 
49.9%  in  five  years. 

(c)  94%  of  the  men  returned  to  work  yesterday. 

(d)  He  saves  25%  of  his  salary. 

{e)  Thompson's  plurality  was  cut  down  37^%  at  the  last  elec- 
tion. 

12.  1/2  =  how  many  per  cent.?    What  is  the  difference  in  per  cent, 
between  1/2  and  1/2%  of  a  quantity? 

13.  Find  the  difference  in  per  cent,  between: 

1/4  and  1/4%.     1/4  and  1/3%.     33i%  and  3/8.     .002  and  1/4%. 

EXERCISE. 
Find  the  missing  values  in  this  table: 

Table  of  Comparative  Value. 
Decimal.  Per  Cent.  Hundredths  (Fraction).  Common  Fraction. 

.0125  

.02  2%  2/100  

.05  1/20 

.061  

12%  

—         m%  —  — 

15%  —  

161%  

20%  

25%  


33i/100 
37i/100 


.80% 


1.75 


90% 


1/2 

11/20 

3/5 

5/8 

2/3 

7/8 
9/4 


PERCENTAGE.  215 

To  Find  the  Percentage. 

The  percentage  of  a  quantity  is  the  value  of  the  hundredths 
taken.  The  base  is  the  number  on  which  the  percentage  is 
reckoned. 

Illustrations.     (1)  I  collect  45%  of  a  debt  of  $720,  or  what  sum? 
Solution.     (Decimal  method.)     45%  =  .45  $720 

.45 
3600 
2880 


$324.00  the  am't  collected. 

(2)  What  profit  is  made  by  selling  a  $42  article  at  3%  advance  on  cost? 
Solution.     (1%  method.)  1%  of  $42  =  $.42 

3%  of  $42  =  1.26  (3X1%  of  $42) 
The  profit  =  3  X  1%  of  $42,  or  $1.26. 

(3)  On  a  $420  purchase,  I  am  allowed  25%  discount,  or  $  ?. 
Solution.     (Aliquot  method.) 

The  discount  is  25%  or  1/4  of  $420.     1/4  of  $420  =  $105,  discount. 
Note.     The  general  tendency  is  toward  the  use  of  the  decimal  method. 

It  is  evident  that  the  percentage  equals  the  product  of  the 
base  by  the  rate. 

FOR    DISCUSSION. 

1.  What  rates  per  cent,  in  the  preceding  table  are  aliquot  parts? 

2.  Is  the  base  concrete  or  abstract? 

3.  Is  the  'percentage  concrete  or  abstract? 

ORAL    DRILL    EXERCISE. 

1.  Fmd  1%  of  24,  36,  500,  4.65,  1/5,  72.5  yd.,  246  lb.  ' 

2.  By  the  1%  method,  find:  3%,  of  2400;  8%  of  20  bbl;  12%  of  $300; 
4%  of  1/3;  15%)  of  .2. 

Note.     100%  of  a  number  =  100/100  of  it,  or  the  number  itself.     200% 
of  a  number  is  how  many  times  the  number? 

3.  Find:  200%  of  2640;  300%  of  $48;  400%  of  1/3;  250%  of  2.4. 

4.  Find:  1/4%  of  5600;  3f%  of  1400;  2/3%  of  1/2;   1/4%  of  1/4; 
3/4%  of  .4. 

5.  Find,  by  aliquot  parts:  25%  of  $3640;  83^%  of  .006;  33i%  of 
1/4;  12^%  of  624  yd.;  90%  of  $7.20. 

6.  Compare  72%  of  $50  with  50%  of  $72.     Conclusion? 

7.  Find:18%of  $250;84%of  75T.;19%of66|A. 


216  BUSINESS  ARITHMETIC. 

EXERCISE. 
Find: 

,      1.    25%  of  27,240  lb.  5.  4^%  of  5/36. 

2.  U%  of  $7654.20.  6.  75%  of  15.02". 

3.  18%  of  .00624.  7.  3/8%  of  $95,240. 

4.  333i%of645T.  8.  62|%  of  11,200  shares. 

EXERCISE. 

Each  problem  illustrates  a  different  use  of  percentage.     State  use  in 
each  case.     Re-word  each  problem  in  question  form.     Find  missing  values. 

1.  The  directors  of  the  Rand&ll  Mfg.  Co.  pass  a  resolution  to  increase 
all  salaries  and  wages  15%.     My  own  salary  of  $1200  is  thereby  raised  to 

$ .     The  advance  will  cause  an  increase  of  $ in  a  monthly  pay 

roll  of  $24,600. 

2.  I  decide  to  set  apart  12^%  of  my  monthly  income  of  $96,   or 
$ ,  for  minor  personal  expenses,  and  to  deposit  in  the  bank,  25%,  or 


3.  On  a  purchase  of  40  O.  C.  stoves,  listed  at  $4.80,  a  customer  is 
allowed  40%  off  the  price,  because  of  the  size  of  his  order.  The  discount 
is  $ and  the  customer  pays  $ each. 

4.  The  purchaser  of  the  stoves  marks  them  at  an  advance  of  25%  on 
what  they  cost  him.     His  selling  price  is  $ ,  of  which  $ is  profit. 

5.  A.  C.  Crane  and  Thomas  Drew  form  a  partnership  with  a  capital  of 
$12,600,  Crane  supplying  33^%,  or  $ ,  and  Drew  %,  or  $ . 

6.  C.  B.  Brown  fails  in  business  and  compromises  with  his  creditors, 
agreeing  to  pay  at  once  66f  %  of  his  debts,  and  a  month  later  15%  more. 
His  debts  total  $46,240.20.     His  payments  are  $ and  $ . 

7.  It  is  reckoned  that  the  machinery  equipment  in  a  factory  decreases 
5%  of  its  original  value  in  each  six  months.  If  worth  $26,400  originally, 
it  is  worth  $ ,  12  months  later. 

8.  An  architect  charges  as  a  fee  for  the  design  of  a  building,  5%  of 
its  cost.     His  fee  on  a  building  costing  $11,380  is  $ . 

9.  Of  700  pupils  in  a  school,  90%,  or pupils,  are  present  on  Janu- 
ary 11.     Of  the  number  present,  40%  are  boys  and are  girls.     In 

an  arithmetic  test,  73%  or pupils  make  a  passing  average. 

10.  "The  newspapers  predict  that  Black's  majority  of  42,400  two 
years  ago  will  be  cut  down  75%  at  the  coming  election.  If  this  is  true 
he  will  still  win  by majority." 


PERCENTAGE.  217 

ORAL    DRILL   EXERCISE. 

(The  teacher  should  prepare  similar  exercises.) 

1.  Base,  240.  Find  1%,  3%,  25%,  33i%,  10%,  11%,  16|%,  87^%, 
1/4%,  20%,  160%,  500%. 

2.  Base  .6.     Find  1/3%,  3/4  of  it,  2/3,  2/3%,  150%,  3%,  400%,  2i%. 

3.  Base,  3/4.     Fmd  x%  of  it,  3%,   20%,   200%,   8%,    12^%,   400%. 

4.  60  decreased  33i%  of  itself  =  ?     (Called  the  difference.) 
320  increased  25%  of  itself  =  ?    (Called  the  amount.) 

5.  Defins  amount  and  difference.  Which  implies  subtraction  and  which 
addition?  If  the  rate  is  46%,  name  the  difference  per  cent,  and  the  amount 
per  cent. 

6.  Decrease.  240  bu.  33|%;  1/4  by  1/3  of  itself;  3/8,  33^%;  2460, 
10%;  800  by  1/4  of  itself;  .6  by  40%;  4.2,  14f %. 

7.  Increase:  12,  33^%;  900,  2/3%;  2172  lb.,  1%;  .04,  125%;  1/4, 
25%;  3.6  T.,  16|% 

In  the  following,  use  each  intermediate  answer  as  the  base  for  the  next 
step. 

8.  Increase  20  by  50%  of  itself;  increase  that  result  33^%;  that 
result,  25%;  that  result  20%;  that  result,  33i%;  decrease  that  result, 
50%;  decrease  that  10%;  increase  10%.     Answer? 

9.  4  increased  50%;  that  result  increased  33^%;  increased  12|%; 
decreased  66f%;  increased  100%;  decreased  50%;  multiplied  by  4; 
decreased  25%;  multiphed  by  1/3.     The  final  result  is  ?. 

Finding  the  Rate  Per  Cent, 
introductory  exercise. 

1.  If  20  =  5%  of  a  given  number,  1%  of  it  equals  ?  and  30=-?%. 

2.  16=  what  part  of  20?    What  per  cent,  of  20? 

3.  1%  of  400=  ?     36=  ?  %  of  400.     924=  ?  %. 
Illustrations.     (1)  What  per  cent,  is  536  of  1200? 
Solution.     (Decimal.)     Since  it  is  desired  to  know  how  many 

hundredths  of  1200  is  536,  divide  536  by  1200 

■44f=44f%. 
1200)536. 
(2)  By  selling  for  $2.27  an  article  costing  $2.00,  what  is  gained  in  per 
cent? 


218  BUSINESS   ARITHMETIC. 

Solution.     (1%  Method.) 

A  gain  of  1%  =$.02.  $2.27  Sell  pr. 

$.27  (total  gain)  divided  by  $.02  equals  2.00  Cost 

number  of  per  cent,  gained.  $.02)$  .27  Gain. 

13|,  or  per  cent,  of 
gain. 

(3)  The  Charleston  basket  ball  team  wins  18  out  of  24  games  or ■ 

%  of  its  gains. 

Solution.     {Fractional.) 

The  question  is:  18  is  how  many  hundredths  of  24? 
18      3       75       ^^^    , 
24  =  4  =  100  =  ^^^«'^"- 

It  is  evident  that  the  rate  equals  the  quotient  of  the  percentage 
divided  by  the  base. 

ORAL    DRILL    EXERCISE. 

Find  missing  values: 

1.  1/4= %  of  1/2,  of  1,  of  3/4,  of  1/8,  of  2,  of  1/16,  of  3/8. 

2.  4.8  = %  of  2.4,  of  4,  of  1.2,  of  9.6,  of  48,  of  6. 

3.  What  per  cent,  of 

2400  is  16?  1.6  is  .4?  4  is  .8?  2/3  is  1/2? 

300  is  29?  .2  is  .06?  7^8  1?  4  is  12? 

4.  $2.40  are %  of  $60.  6  qt.  are %  of  20  qt. 

760  lb  are %  of  300  lb.         90  sq.  yd.  are %  of  2000  sq.  yd. 

5.  What  per  cent,  of 

3  is  4?  25  is  30?  2.5  is  5?  1/4  is  1/3? 

4  is  3?  25  is  5?  .25  is  .5?  3/8  is  5/8? 

6.  4  is  what  per  cent,  of  5?  5  of  3?  3  of  4?  4  of  6?  6  of  4?  6  of  8? 
8  of  6?  8  of  10?  10  of  15?  15  of  20?  20,  mcreased  20  %,  of  30?  30, 
decreased  1/5,  of  40? 

GENERAL    EXERCISE. 

1.  What  per  cent  of  the  cost  of  a  $160  horse  is  lost  by  selling  it  for  $120? 

2.  I  invest  $6000  in  a  partnership  capitalized  at  $27,000,  under  an 
agreement  to  share  in  profits  according  to  investment.  What  per  cent. 
of  any  profits  should  I  receive 

3.  If  $64  is  the  cost  of  the  premium  on  a  $4000  fire  insurance  policy, 
what  is  the  rate  of  premium? 

4.  What  per  cent,  has  a  city's  population  increased  in  a  year,  if  it  now 
has  240,000  inhabitants,  and  a  year  ago  had  220,000? 


PERCENTAGE.  219 

5.  How  does  a  profit  of  $4  on  a  $20  investment  compare  in  rate  with  a 
profit  of  $16  on  a  $60  investment? 

6.  What  per  cent,  is  made  in  a  class  test  if  943  credits  are  obtained 
out  of  a  possible  1140? 

7.  A  new  printing  press  has  a  maximum  output  of  35,000  copies  per 
hour,  compared  with  an  output,  for  the  machine  it  replaced,  of  27,500. 
What  is  the  per  cent,  of  increase? 

Note.  Percentage  is  frequently  employed  in  technical  writing  and 
statistics,  to  show  proportions,  rates  of  increase  and  decrease  and  for 
comparisons  of  values. 

8.  Find  the  missing  rates  in  this  table  to  three  places  decimally: 

Where  Exports  Find  Markets. 


Countries. 

Total  Imports. 

Share  from  U.  S. 

Per  Cent. 

United  Kingdom, 

$2,958,289,000. 

$638,006,000. 

Germany, 

1,696,660,000. 

236,082,000. 

Italy, 

398,463,000. 

45,956,000. 

China, 

349,913,000. 

36,304,000. 

Canada, 

290,361,000. 

175,862,000. 

Australia, 

217,676,000. 

22,549,000. 

Japan, 

.208,554,000. 

34,834,000 . 

Spain, 

204,401,000. 

23,006,000. 

Argentina, 

197,974,000. 

27,908,000. 

Denmark, 

166,837,000. 

26,832,000. 

Brazil, 

161,587,000. 

18,518,000. 

Mexico, 

109,884,000. 

72,509,000. 

ORIGINAL    EXERCISE. 
Write  and  solve  original  problems  illustrating  the  use  of  rates  per  cent.: 

1.  In  measuring  the  increase  of  sales  of  a  business  during  one  year. 

2.  In  measuring  the  loss  of  a  firm  through  bad  debts.  , 

3.  In  comparing  the  expense  of  repairs  on  a  machine  with  its  original 
cost. 

4.  In  comparing  the  votes  of  successful  and  defeated  candidates  for 
office,  in  a  political  election. 

EXERCISE. 
Compute  missing  values  in  this  article. 

BASEBALL  AVERAGES. 

Baseball  records  furnish  an  excellent  illustration  of   the  use  of   rates 
per  cent,  in  showing  relative  standing.    Thus,  the  standing  of  any  team  is 


220  BUSINESS   ARITHMETIC. 

determined  by  the  per  cent,  of  games  won  out  of  games  played.  This 
rate  is  computed  to  three  places  decimally.  For  example,  the  standing  of 
the  American  League  on  Sisptember  25,  of  a  recent  year  was: 

American  League  Standing. 

Team.             Won.     Lost.     P.  C.             Team.          Won.  Lost.  P.  C. 

Philadelphia. .  .84          51          .622         Detroit 70  69         .504 

Chicago 82          54            ?            New  York 66  66          ? 

Boston 69          66             ?            Washington ...  56  80          ? 

Cleveland 70          69            ?             St.  Louis 48  90          ? 

Owing  to  bad  weather,  or  other  causes,  some  scheduled  games  may  be 
postponed,  so  that  on  any  particular  date,  the  number  of  "games  played" 
by  each  team  may  vary.  (Is  this  true  above?)  It  is  evident  that  per 
cent,  rates  offer  a  better  means  of  comparison  than  common  fractions. 
(Show  why.) 

Of  course,  averages  change  every  day  when  teams  are  playing.  On 
Sept.  26,  Washington  added  a  game  to  its  "won"  column  by  defeating 
Chicago,  whose column  was  increased  by  1.  Detroit  lost  to  Phila- 
delphia; New  York  defeated  St.  Louis;  and  Cleveland  beat  Boston.  At 
the  close  of  September  26,  the  standing  would  appear  as  follows: 

American  League  Standing. 

Team.            Won.     Lost.  .  P.  C.            Team.          Won.     Lost.     P.  C. 
Philadelphia.  .85  51  ?  —         —         — 


Towards  the  close  of  the  season  possible  results  are  often  forecasted. 
On  Sept.  26,  for  example,  these  possibilities  might  be  considered: 

1.  Chicago  and  Philadelphia  have  each  eight  more  games  to  play. 
If  Chicago  wins  six  and  loses  two,  while  Philadelphia  wins  four  and  loses 
four,  will  Chicago  be  in  the  lead? 

2.  Philadelphia  plays  the  first  four  of  these  games  in  Chicago.  If 
Chicago  should  win  three  would  Philadelphia  lose  the  lead? 

3.  A  Philadelphia  enthusiast  asks  "  How  many  of  these  last  eight  games 
must  Philadelphia  win  to  be  sure  of  the  championship?  " 

In  a  similar  way  the  records  of  pitchers  are  determined.  Thus  a  pitcher 
winning  25  games  and  losing  18  would  have  an  average  of  games  won  of  . — . 

In  batting,  the  player  who  makes  93  safe  hits  in  240  times  at  bat  is 
said  to  have  a  batting  average  of  . — . 


PERCENTAGE.  221 

The  average  of  the  players  "in  the  field"  are  based  on  three  factors: 
(1)  the  number  of  opposing  players  they  "put  out";  (2)'  the  number  of 
plays  in  which  they  "assist"  in  putting  out  a  player;  (3)  any  attempted 
plays  in  which  the  player  fails  to  complete  a  play  within  his  power  (errors). 
The  players'  fielding  averages  are  expressed  decimally  as  the  per  cent.  * 
that  (1)  plus  (2)  is  of  the  sum  of  (1),  (2)  and  (3). 

Fielding  Averages. 


Player. 

Outs. 

Assists. 

Errors 

Percent. 

C.  Baker  (Outfielder) 

217 

109 

21 

? 

J.  Town  (1st  base) 

824 

103 

37 

? 

S.  Crane  (2d  base) 

162 

219 

33 

? 

T.  Parker  (Shortstop) 

124 

137 

41 

? 

FINDING  THE  BASE. 
INTRODUCTORY    EXERCISE. 

1.  If  14  =  2%  of  a  number,  1%  =  ?;  100%  or  the  number  =  ? 

2.  Find  100%,  if  2%  =4,  32,  420,  6. 

3.  Find  the  numbers  of  which  33i%  =35,  1240,  520,  9,  1/4. 

4.  Find  100%,  if  300%  =  700,  45,  6. 

Illustrations.     (1)  What  is  the  cost  of  an  article  that  sells  for  $618, 
at  3%  profit? 

Analysis  and  Solution.     Let  the  cost  equal  100%. 
The  selling  price  equals  103%  of  the  cost  and  the  cost  equals  100/103 
of  $618.  $600,  cost. 

1.03)$618 
(2)  40  pounds  of  Mocha  coffee  will  form  66 f%  of  a  mixture  of  how 
many  pounds  of  Mocha  and  Java? 

Solution.    40  lb.  =  66f  %  or  2/3         40  =  2/3  of  mixture, 
of  the  mixture.  20  =  1/3  of  mixture. 

60  =  3/3  of  mixture,  or  the  total 
quantity. 

It  is  evident  that  the  base  equals  the  percentage  divided 
by  the  rate. 

ORAL    DRILL    EXERCISE. 

1.     When  is  the  "1%  method"  superior  to  the  aliquot  or  fractional 
method?     Which  is  better  in  the  following? 
40  =  10%  of  ?;  27=9%  of  ?;  62=50%  of  ?;  28=4%  of  ?;  75=33i  of  ? 


222  BUSINESS   ARITHMETIC. 

•   2.  24  =4%  of?;  25%  of?;  .4%  of?;  1/3%  of?;  150%  of? 

3.  420  =  6%  of  ?;  250%  of  ?;  40%  of  ?;  25%  of  ?;  1/4%  of  ? 

4.  2/3=25%  of  ?;  3%  of  ?;  10%  of  ?;  400%  of  ?;  1/2%  of  ? 

5.  426  =  25%  of  ?;  6750  =  50%  of  ?;  .06  =  75%  of  ?;  1/4  =  800%  of  ? 

6.  16  =  1/4%  of  ?;  480  =  2/3%  of  ?;  12,000  =  U%  of  ?  .006  =  4/3%  of  ? 

EXERCISE. 

Find  missing  values: 

1.  297.6  =  3/5%  of  — .  4.     45.297  =  7%  of  — . 

2.  97,485  =  20%  of  — .  5.     1605  =  120%  of  — . 

3.  2.0406  =  1/8%  of  — .  6.     29,623  =  47.2%  of  — . 

EXERCISE. 

(Each  problem  illustrates  a  practical  use  of  this  phase  of  percentage.) 
Find  missing  values. 

1.  What  sum  must  I  invest  at  a  profit  of  7^%  per  year  to  earn  a 
profit  of  $900  yearly? 

2.  A  certain  cloth  shrinks  6^%  in  dyeing.  In  order  to  have  640  yd. 
of  finished  material,  one  must  dye  —  yd. 

3.  "Thus  the  price  has  fallen  16%  in  two  months,  and  the  money  that 
would  buy  100  shares  two  months  ago,  will  now  buy  —  shares." 

4.  A  merchant  sends  a  manufacturer  an  order  for  600  articles,  enclosing 
a  draft  in  payment,  and  writing,  "In  case  you  grant  me  a  discount  from 
your  last  quotation,  send  all  the  goods  the  money  will  buy."  The  manu- 
facturer allows  a  discount  of  20%  from  his  last  quotation.  What  number  of 
articles  should  he  ship  his  customer? 

(Note:  " grant  a  discount "  means  "lower  the  price.") 

5.  For  a  certain  contract  a  carpenter  requires  72,000  ft.  of  flooring. 
As  he  knows  the  waste  in  cutting  will  average  25%,  he  should  order  —  ft. 

6.  If  a  farmer  ofifers  to  sell  his  land  "at  75%  of  its  real  value"  for 
$8250 — in  order  to  dispose  of  it  at  once — he  places  what  valuation  on  it? 

ORAL    DRILL   EXERCISE. 

All  Cases. 
Find  missing  values: 

1.  16=4%  of  — ;  that  result =20%  of  — ;  that  result  =200%  of  — ; 
that  =  500%  of  — ;  that  =20  times  — ;  that =25%  of  — ;  that  result  =400% 
of  — ;  that  =  l/2%of  — . 


PERCENTAGE. 


223 


2. 


A  percentage  study  of  36. 

(a) 

(&) 

(c) 

1%  1 

"     4%of  — 

r  1200 

5% 

100%  of  — 

200 

20% 

80%  of  — 

3000 

150% 

7%  of  — 

30 

400% 

►of  36^ ;36  =  - 

3%  of—  36  =- 

-  %  of  - 

6 

33^% 

33i%  of  — 

1/2 

1/2% 

2/3  of  — 

3.6 

2  3% 

2i%  of  - 

.9 

U% 

L  U%of-    • 

40000 

EXERCISE. 

1.  121%  of  372  =  li%  of  — . 

2.  8%  of  33i%  of  960  =  — %  of  1200. 

3.  175%  of  36  =  1/3  of  24%  of  — . 


EXERCISE. 

An  Argument  for  the  Panama  Canal. 


Trip. 


New  York  to  San  Francisco  via  Panama,  5278  M. ;  via  Cape  Horn,  13,340  M. 
San  Francisco  to  Liverpool  via  Panama,.  7907  M.;  via  Cape  Horn,  13,678  M. 
New  York  to  Manila 

via  Panama  11,412  M.;  via  Cape  Good  Hope,  13,555  M. 
New  York  to  Yokohama 

via  Panama  9692  M.;  via  Cape  Good  Hope,  15,178  M. 
New  York  to  Sydney 

■  via  Panama,  9560  M.;  via  Cape  Good  Hope,  12,218  M. 

Present  these  figures  in  tabular  form.  Express  also,  the  distance  via 
Panama,  (o)  as  a  per  cent,  of  other  routes;  (6)  via  the  other  routes  as  a 
per  cent,  of  the  Panama  route. 


ANALYSIS  DRILL. 
MIXTURES  AND  COMPOUNDS. 

Oral. 

1.  20  lb.  of  one  grade  of  tea  are  blended  with  30  lb.  of  a  second  grade, 
forming  a  mixture  of  ?  lb.,  of  which  ?%  is  first  grade.  The  quantity  of  first 
grade  is  ?  %  of  that  of  second  grade. 

2.  30  lb.  are  blended  in  the  ratio  of  5  :  4  with  ?  lb.  of  a  second  grade. 
The  second  grade  forms  ?  %  of  the  mixture  and  equals  in  quantity  ?  % 
of  the  first  grade. 


224  BUSINESS   ARITHMETIC. 

3.  How  much  witch  hazel  is  required  for  8  gal.  of  a  12%  solution? 

4.  ?  lb.  Mocha  and  ?  lb.  Java  are  required  for  a  mixture  of  180  lb., 
if  the  quantity  of  Mocha  exceeds  that  of  Java  25%. 

Written. 

1.  What  quantity  of  each  ingredient  is  required  for  25  gal.  of  a  mixture 
of  hard  oil  diluted  40%  with  turpentine? 

2.  A  mixture  of  tallow  and  rosin  is  to  contain  15%  tallow.  How  much 
tallow  is  required  for  40  lb.  of  rosin? 

3.  22  lb.  of  standard  bulhon,  80%  pure,  will  make  —  lb.  of  standard 
purity  (90%). 

4.  The  products  of  the  distillation  of  100  lb.  of  average  gas  coal  of  a 
certain  mine  approximate:  64  lb.  coke,  6|  lb.  tar,  12  lb.  ammonia  liquor, 
14  lb.  purified  gas,  and  —  lb.  minor  products  and  waste.  Express  com- 
position in  per  cents.  What  amount  of  coal  (2240  lb.  to  the  ton)  must  be 
distilled  to  manufacture  1,500,000  cu.  ft.  of  gas,  if  gas  measures  10,540  cu.  ft. 
to  the  ton? 

5.  The  composition  of  Babbitt's  metal  is  3.7%  copper,  89%  zinc,  and 
7.3%  antimony.  Determine  the  quantity  of  each  metal  required  for  an 
order  of  16  T. 

6.  The  analysis  of  a  metallic  paint,  per  5000  parts,  is:  3943  parts  iron 
peroxide,  165  parts  alumina,  598  parts  silica,  254  parts  water,  16  parts 
lime,  14  parts  manganese,  —  parts  minor  elements.  Express  the  paint 
formula  in  per  cent.  form. 

7.  A  common  asphalt  varnish  contains  3,  4  and  18  parts  respectively 
of  asphaltum,  boiled  oil  and  turpentine.  Give  formula  in  percentage  form. 
What  quantity  of  asphaltum  is  required  to  combine  with  600  gal.  of  the 
proper  mixture  of  oil  and  turpentine? 

8.  A  mixture  of  three  grades  of  material  prepared  in  proportions  of 
20%,  30%  and  50%,  is  found  to  be  markedly  improved  by  reversing  the 
proportions  of  first  and  second  grades.  It  is  decided  to  change  120  lb. 
of  the  old  mixture  to  the  new  by  adding  whatever  quantities  are  necessary. 
What  quantities  should  be  added?  What  is  the  quantity  of  the  new 
mixture? 


CHAPTER   XXX. 

ELEMENTARY  PROFIT  AND  LOSS. 

INTRODUCTORY. 

1.  What  do  I  gain  by  selling  an  article  that  costs  me  $400  at  a  price 
25%  higher? 

2.  If  I  sell  for  $40  a  desk  that  cost  me  $60  what  do  I  lose?  What  per 
cent,  of  the  cost  do  I  lose? 

3.  Compare  these  terms  with  the  corresponding  terms  of  percentage: 
Cost,  selling  price  at  a  loss,  selling  price  at  a  profit,  rate  of  gain  or  loss. 

Practically  all  transactions  in  business  are  carried  on 
directly  or  indirectly  for  profit,  or  to  prevent  or  reduce  loss, 
or  to  decrease  expenditure.  A  business  man  continually  faces 
the  question,  "Does  it  pay?'*  The  question  may  concern  the 
selling  price  of  an  article,  the  results  of  some  business  policy, 
the  comparison  of  investments,  or  endless  other  conditions. 
In  most  cases  arithmetic  is  used  to  determine  the  answer. 
Percentage  is  very  commonly  used  to  measure  results  where 
profit  may  be  expressed  in  money. 

ILLUSTRATIVE    EXERCISE. 

Find  missing  values: 

1.  Tables  bought  at  $8  are  sold  at  $10,  at  a  profit  of  — %  on  cost. 

2.  I  sell  apples  costing  $2  per  barrel  at  $3  and  dispose  of  6  bbl.  per  day. 
By  selling  at  $2.50  I  find  my  sales  average  20  bbl,  per  day.  Lowering  my 
selling  price  increases  my  profits  $ —  or  — %  per  day. 

3.  By  the  installation  of  new  machinery',  a  manufacturer  reduces  the 
force  required  for  the  same  output  from  500  to  420,  thus  affecting  a  re- 
duction of  —  %. 

4.  A  manufacturer,  by  a  change  of  process  reduces  the  cost  of  manu- 
facture of  a  rug  from  $4.00  to  $3.80.  If  his  sellmg  price  remains  $4.80 
his  profits  increase  —  %. 

16  225 


226  BUSINESS  ARITHMETIC. 

Note.  The  principle  of  projit  as  a  saving  influences  our  daily  acts  as 
individuals. 

5.  I  can  buy  potatoes  at  an  up-town  store  for  $3.00  per  bbl.,  and  at  a 
down-town  market  for  $2.65.     By  buying  at  the  latter,  I  save  —  c. 

6.  I  can  buy  a  working  bench  for  $16.00  or  I  can  buy  material  for  $4.00 
and  make  it  myself,  at  a  saving  of  $  — ,  or  —  %.  If,  however,  my  time 
is  worth  $1.50  an  hour  for  the  four  hours  required,  I  save  only  $ — . 

Ordinarily,  'profit  or  loss  is  reckoned  as  a  percentage  of  cost. 

The  cost,  however,  is  an  indefinite  quantity.     In  addition  to 

what  he  pays  the  manufacturer  for  goods,  the  merchant  pays 

freight,  insurance  and  storage,  the  expense  of  sale,  and  the 

cost  of  keeping  bookkeeping  records,  etc.     Thus  the  question, 

often  arises  as  to  what  incidental  charges  shall  be  added  to 

first  cost,  or  subtracted  from  selling  price  to  determine  a  base. 

Partly  to  avoid  this  difficulty,  some  dealers  reckon  individual 

profits  as  a  per  cent,  of  sales. 

Illustrations.     (1)  I  buy  6  desks   @  $31  each,  paying  $6  freight. 
Find  the  selling  price  each  at  25%  profit. 
Solution.     Net  cost  as  hose. 

15%  or  1/4  of  $31  =$7.75  profit.     $31'+$7.75 =$38.75  selling  price 
+$1.00  freight  =  $39.75,  gross  selling  price. 
Solution.     Gross  cost  as  base. 
The  freight  is  $1.00  per  desk. 
$31 +$1  =$32,  gross  cost  per  desk. 
5/4  of  $32  =  $40,  sell,  price. 

(2)  One  desk  is  sold  for  $36  or  ?  %  profit. 

Solution.     Net  cost  as  base.     The  profit,  $36— $31  =$5. 

The  rate,  5/31,  or  16.1%. 
Gross  cost  as  base.      The  profit,  $36-  $32  =  $4. 

The  rate,  4/32,  or  12.5%. 
Sell,  price  as  base.       The  profit,  $36  -  $32  =$4. 

The  rate,  4/36,  or  11.1%. 
Note.     In  this  hook   profit  ivill  be  reckoned  as  a   per  cent,  of  net  cost 
unless  otherwise  stated.     Read  problems  carefully. 

The  department  store  offers  a  good  illustration  of  profit 
reckoning.  Here  the  selling  price  is  fixed  by  the  merchandise 
manager.,  or  by  the  department  buyer.  Computations  are  on 
a  percentage  basis.     The  rate  of  profit,  however,  varies  with 


ELEMENTARY  PROFIT  AND   LOSS.  227 

the  article  or  department.  Rapid  selling  articles  yield  a  lower 
rate  than  slow  selling  goods.  The  latter  may  sell  at  30% 
profit,  where  the  rapid  seller  brings  5%.  The  department, 
however,  that  does  $60,000  worth  of  business  annually  on 
$10,000  capital  can  deal  on  a  smaller  margin  of  profit  than  one 
that  does  $80,000  business  on  $15,000  capital.  By  the  more 
rapid  "turning  over"  of  capital,  goods  sold  at  low  profits  may 
bring  a  department  as  high  an  average  as  others  sold  at  a  high 
rate.  Suppose  an  article  sold  a  month  after  purchase  at  5% 
profit,  the  original  cost  being  reinvested  in  the  same  goods, 
and  the  process  continued  for  a  year.  The  profit  on  original 
capital  is  then  5  X  12,  or  60%  for  the  year.  The  tendency 
is  to  make  all  departments  yield  the  same  rate  annually  by 
means  of  higher  marking  of  slow  selling  goods. 

To  the  price  at  which  the  goods  are  billed  at  the  store,  are 
added  the  fixed  charges  for  rent,  bookkeeping,  selling,  delivery, 
etc.  These  have  been  reduced  to  an  average  per  cent,  of  cost 
for  each  store  or  department — ranging  from  15%  to  25%. 
Many  of  the  stores  reckon  the  profit  as  a  per  cent,  of  sales. 
It  is  then  a  simple  matter  to  approximate  any  day's  profits 
from  the  amount  of  sales. 

I.    Gains,  Losses  and  Selling  Prices. 

EXERCISE. 

Illustrating  simple  phases  of  profit  and  loss. 
Answer  orally  if  possible. 

1.  Find  the  loss  on  desks  costing  $25  if  sold  at  10%  loss;  at  20%  loss; 
at  12^%  loss;  at  5%  loss;  at  40%  loss. 

2.  Find  the  gain  on  toys  costing  $1.60  and  sold  at  25%  gain;  at  20% 
gain;  at  5%  gain;  at  15%  gain. 

3.  Find  the  gain  or  loss  if  goods  cost  $1.20  a  doz.  and  are  sold  at  15  c 
each;  at  20  c  each;  at  $2.00  per  doz.;  at  5%  profit;  at  20%  loss;  at  $1.05 
per  doz. 

4.  From  what  known  values  may  the  following  be  computed:  (a) 
Selling  price  at  gain,  (6)  gain,  (c)  selling  price  at  loss,  (d)  loss? 


228  BUSINESS  ARITHMETIC. 

5.     Find  the  missing  values  by  inspection. 

Cost.         Rate  Gain  or  Loss.     Selling  Price.      Gain  or  Loss. 

a.  $56.  20%  gain  $ $ 

6.  9.60  16f%loss. 

c.  4260.00  25%  gain 

d.  3.88  doz.  

e.  1.80  % 39  c 

EXERCISE. 

1.  What  profit  is  realized  by  buying  gloves  at  $9.20  per  doz.  and  selling 
them  at  $1.15  per  pair? 

2.  The  goods  in  the  crockery  department  of  a  store  are  marked  at  20% 
profit  and  18%  store  expense.  What  is  the  selling  price  of  plates  costing 
$9.60  per  doz.? 

3.  The  book  buyer  of  a  department  store  pays  $7.50  for  ten  copies  of  a 
popular  novel  which  he  disposes  of  at  $1.08.  As  each  lot  is  disposed  of 
he  re-invests  the  original  sum  in  a  new  lot,  until  he  has  disposed  of  120 
copies.    He  has  "  turned  over  "  his  capital  —  times,  and  gained  a  total  of  $ — . 

4.  Apples  are  bought  ©  $2.40  per  bbl.  (3  bu.)  and  are  retailed  at  90  c. 
per  bu.  Determine  the  final  loss  or  gain,  on  a  lot  of  60  bbl.,  the  average 
loss  by  decay  being  15%. 

5.  A  dealer  retails  at  $6.50  a  grade  of  flour  that  costs  him  $5.00  per 
bbl.  The  wholesale  price  is  raised  10%,  and  the  dealer  immediately 
raises  his  price  to  $7.50,  but  his  sales  fall  off  10%  in  amount.  How  does 
this  affect  his  net  profits? 

II.    Profit  as'  a  Rate  of  Selling  Price. 

(Omit  if  desired.) 
ORAL    EXERCISE. 

1.  If  30%  of  selling  price  is  profit,  how  many  per  cent,  is  cost? 

2.  If  $14  is  the  cost,  in  ex.  1,  find  the  selling  price  and  profit. 

3.  The  goods  in  a  grocery  department  are  sold  at  a  profit  of  15%  on 
selling  price.     What  are  the  profits  on  sales  of  $6400? 

4.  Find  the  selling  price  of  a  table  costing  $6.00  to  gain  25%.  To 
gain  33i%;  to  lose  20%;  to  lose  $1.40;  to  gain  20%. 

EXERCISE. 

1.  Find  the  selling  price  of  desks  costing  $16.00,  to  allow  for  20% 
store  burden  (based  on  cost)  and  25%  profit. 


ELEMENTARY   PROFIT   AND   LOSS.  229 

2.  Find  the  profit  on  an  invoice  of  $2400  worth  of  goods,  sold  at  20% 
profit,  the  expense  of  sale  being  $98.60. 

3.  Find  the  selling  price  of  silks  costing  $1.80  per  yd.,  if  store  burden 
is  20%,  profit  allowance  20%,  waste  in  cutting  5%. 

4.  Which  base  for  profit,  cost  or  selling  price,  yields  the  higher  profit 
at  any  given  rate?     Illustrate. 

III.    Per  Cents  of  Gain  or  Loss. 

ORAL    EXERCISE. 

1.  By  selling  a  $2.00  engraving  for  $2.40  a  profit  of  —  c  or  — %  of 
the  cost  is  realized. 

2.  A  profit  of  75  c  on  a  $5.00  article  is  a  profit  of  — %. 

3.  A  carriage-maker  buys  a  cart  for  $40.00,  expends  $15  in  repairs 
and  sells  it  for  $69.00,  making  — %  profit. 

4.  .Does  it  pay  a  dealer  to  reduce  his  profit  from  15%  to  9%,  if  his 
sales  are  thereby  doubled? 

5.  Compute  the  rate  of  gain  or  loss. 

Cost.  Selling  Price, 

(a)  $3.20  $3.60  (d) 

(6)  4.00  3.98  (e) 

(c)  8.40  9.60  (/) 

Selling  Price  Gain. 

(jg)  $8.00  $2.00  (i) 

(h)  15.00  5.00  (/c) 

7.    Find  the  rates  in  g-k,  ex.  5,  if  based  on  selling  price. 

EXERCISE. 

1.  Skates  purchased  at  $11.40  per  doz.  are  retailed  at  $1.60  per  pair, 
yielding  a  profit  of  — %. 

2.  A  real  estate  dealer  bought  a  house  at  auction  for  $3600.  He 
expended  $160.00  in  repairs,  $95.00  in  painting,  $5.00  for  advertising  and 
then  sold  the  property  for  $4250.  Compute  his  gain  per  cent,  on  invest- 
ment. 

3.  A  dealer  purchased  60  doz.  straw  hats  @  $9.00  a  doz.  He  sold 
42  doz.  @  40%  profit,  and  6  doz.  @  $1.25  each.  If  he  disposes  of  the  balance 
at  cost,  what  is  his  net  rate  of  gain  or  loss? 

4.  A  novelty,  manufactured  at  a  cost  of  40c,  is  to  be  retailed  at  a  price 
to  yield  manufacturer,  jobber,  wholesaler  and  retailer  each  a  profit  of  10c. 
What  rate  of  profit  is  realized  by  each? 


Cost. 

Gain. 

Loss. 

$600 

$  .90 

15.00 

1.20 

4000.00 

$180.00 

jlling  Price. 

Loss. 

$  12.00 

$    3.00 

150.00 

30.00 

230  BUSINESS   ARITHMETIC. 

5.  A  dealer  sells  Mocha  coffee,  costing  25c,  @  32c;  and  Java,  costing 
28c,  @  36c.  A  mixture  of  the  two  grades  containing  one-third  Mocha, 
readily  sells  @  40c.     Compare  rates  of  gain  on  straight  grades  and  blends. 

6.  If  a  dealer  sells  at  30%  below  his  regular  price  of  30%  profit,  he 
loses  what  per  cent,  of  profit? 

IV.    The  Cost. 

(Answer  orally  if  possible.) 

1.  By  selling  a  coat  for  $6.00  a  dealer  gains  20%  on  cost.  What  is 
the  cost?     Suggestion:  $6.00  =  120%  of  cost. 

2.  If  a  profit  of  45c  each  is  realized  on  hats  sold  for  $1.65  what  do  they 
cost  per  dozen? 

3.  $40.00  represents  5%  profit  on  what  sum?     25%  loss  on  what  sum? 

4.  Find  the  cost  from  the  values  given: 


Selling  Price.  Gain.  Selling  Price. 

(a)  $     5.40  $1.40  (c)       $     2.50  $.18 

(6)  $240  25%  (d)       $960.  33  1/3% 

Gain.  Rate.  '      Loss.  Rate. 

(e)  $     .84  7%  (g)        $     .52  4% 

(/)  $30.  20%  (h)       $84.  25% 

5.  Markward  offers  to  sell  a  piece  of  land  at  "40%  below  cost,"  for 
$750.     He  evidently  values  the  land  at  what  price? 

6.  If  the  retail  price  of  an  article  is  fixed  by  competition  at  $5.40  the 
cost  of  manufacture  must  be  kept  down  to  what  figure  to  allow  for  15% 
profit  to  the  manufacturer  and  20%  to  the  dealer? 

Examples  in  Savings. 

7.  A  manufacturer,  by  a  change  in  process,  reduces  the  cost  of  office 
desks,  which  he  sells  direct,  for  $30.00,  from  $21.00  to  $18.60.  The  saving 
in  cost  is  ?  %  and  the  increase  in  profits  ?  %. 

8.  By  the  installation  of  new  machinery,  the  time  required  for  certain 
work  was  reduced  from  17  hours  to  14|  hours.  If  labor  costs  48c  per 
hour  this  represents  what  saving  on  each  piece  of  work? 

9.  By  a  change  in  the  boiler  equipment  of  a  factory,  the  coal  con- 
sumption was  reduced  from  34  tons  per  day  to  31i  tons,  or  ?  %. 

GENERAL    EXERCISE. 

1.  Does  it  pay  to  buy  apples  @  $2.40  per  bbl.  (2J  bu.)  and  to  sell 
them  @  32c  per  pk.,  if  10%  is  allowed  for  decay? 


ELEMENTARY   PROFIT   AND   LOSS.  231 

2.  What  rate  of  profit  is  obtained  by  retailing  eight  articles  at  the 
price  they  cost  per  dozen? 

3.  A  blend  of  32c  and  18c  coffee,  mixed  in  the  ratio  of  3  to  4,  is  to  be 
put  on  the  market  in  two-pound  packages,  and  sold  at  25%  advance  on 
cost.     Find  the  selling  price  per  package. 

4.  Does  it  pay  to  lower  the  selling  price  of  tea  (costmg  48c)  from 
80c  to  72c,  if  the  change  in  price  leads  to  a  10%  increase  in  amount  of  sales? 

5.  What  is  the  per  cent,  of  gain  on  cost  if  16%  of  what  is  received  for 
an  article  is  gain? 

6.  What  retail  selling  price,  on  an  article  costing  $8.00  to  manufacture, 
will  yield  profits  of  25%  to  the  retailer  and  20%  to  the  wholesaler  and  to 
the  manufacturer? 

7.  At  a  price  yielding  6%  profit,  the  sales  of  a  certain  article  equal 
3600 pkg.  per  week;  at  a  price  yielding  15%  profit,  sales  average  2500  pkg. 
per  week.     Which  is  the  more  profitable  price  and  why? 

8.  A  60c  grade  of  pepper  is  blended  with  a  30c  grade  in  the  ratio  of 
3  to  5,  and  the  blend  is  retailed  at  50c  By  increasing  the  proportion  of 
the  fiirst  grade  to  50%,  sales  increase  20%.  Does  the  change  of  proportion 
pay,  if  there  is  no  increase  in  selling  price? 

ORAL    DRILL    EXERCISE. 
I. 

The  cost  of  a  certain  make  of  chair  is  $6.00. 

1.  What  is  the  selling  price  each  at  10%  gain?  (Give  analysis.)  At 
5%  loss?  At  8%  gain?  At  20%  loss?  At  15%  profit?  At  a  profit  of 
36c?  At  a  profit  of  $1.62  for  three  chairs  (analysis)?  What  is  the  selling 
price  of  two  chairs  at  20%  gain?     Of  half  a  dozen  at  20%  loss? 

2.  What  is  the  selling  price  of  one  chair  if  20%  of  the  selHng  price  is 
profit?  If  25%  of  the  selling  price  is  loss?  If  $1.40  is  loss?  If  87  cents 
is  gain? 

3.  What  is  gained  by  selling  one  chair  at  5%  profit  (analysis)?  On 
three  at  10%  profit?  On  one  dozen  at  20%  profit?  On  three  sold  for 
$20.60?  What  is  lost  on  one  at  5%?  On  six  sold  at  12|%  below  cost? 
By  selling  three  for  $17.00?  What  is  the  profit  if  20%  of  the  selling  price 
is  profit?     (Analysis.) 

4.  What  per  cent,  is  gained  by  selling  one  chair  for  $6.50?  Three  for 
$24  00?  Two  for  the  cost  6f  three?  Five  for  what  one-half  dozen  cost? 
What  per  cent,  is  lost  by  selling  at  18c  below  cost?  At  $4.73?  By  selling 
three  for  $15.00?    By  selling  three  for  the  cost  of  two? 

What  is  the  rate  of  gain  or  loss,  based  on  selling  price,  on  one  chair 


232  BUSINESS   ARITHMETIC. 

sold  for  $7.20.  (Give  analysis.)  On  three  sold  for  $21.00?  On  two  sold 
for  $10.00.     On  four  sold  at  price  equal  to  the  cost  of  three? 

5.     The  selling  price  of  an  article  is  $3.60. 

The  cost  must  be  limited  to  $  ?  to  gain  20%?  To  gain  25%  based 
on  selling  price?  To  gain  40%.  To  lose  $1.23?  What  per  cent,  of  profit 
on  cost  is  realized  by  buying  at  $2.40?    At  $3.00?    At  $3.20? 

II. 
Give  and  solve  an  origmal  example  to  illustrate  each  case. 

1.  From  what  known  values  may  cost  be  computed?     (Several  cases.) 

2.  From  what  known  values  may  selling  price  be  computed?  Rate  of 
loss?     Rate  of  gain?     Gain?     Loss? 

3.  What  values  can  be  computed  from  cost  and  loss?  From  gain  and 
rate?     From  selling  price  and  rate?     From  cost  and  selling  price? 

4.  Cost  and  ?  determine  rate  of  gain. 

5.  Selling  price  and  ?  determine  rate  of  profit 

FOR    DISCUSSION    AND    ILLUSTRATION. 

1.  Why  may  a  dealer  lose,  who  sells  an  article  for  the  price  he  paid  for 
it? 

2.  Could  a  dealer  lose  if  he  sold  an  article  for  more  than  he  paid  for  it? 

3.  What  may  increase  the  first  or  direct  cost  of  an  article? 

4.  Why  may  a  dealer  have  less  of  a  certain  lot  of  mdse  to  sell  than  he 
bought? 

5.  How  may  the  following  affect  the  selling  price  of  an  article:  over- 
stocking; competition;  style;  fire;  depreciation;  season  of  the  year,  locality? 

6.  What  may  lead  to  the  sale  of  parts  of  a  lot  of  mdse.  at  different 
rates? 

7.  What  is  meant  by  "quick  sales  and  small  profits"? 

8.  May  sales  at  a  fixed  price  increase,  and  yet  profits  decrease? 

9.  Under  what  circumstances  may  profits  increase,  if  selling  price 
decreases? 

10.  Under  what  circumstances  may  profits  decrease,  if  the  selling  price 
increases? 

MARKING  GOODS. 

In  many  businesses  it  is  a  common  custom  to  mark  goods, 
directly  or  by  means  of  a  tag,  with  the  selling  price  and,  at 
times,  with  the  cost.     The  cost,  and  sometimes  the  selling 


ELEMENTARY   PROFIT   AND   LOSS.  233 

price,  is  written  in  a  key  or  private  symbol,  so  that  it  may  be 
known  only  to  certain  employees  of  the  business.  While 
some  keys  are  exceedingly  complicated,  consisting  of  arbitrary 
characters,  the  usual  form  consists  of  a  word  or  phrase  con- 
taining ten  different  letters,  each  of  which  stands  for  a  par- 
ticular figure.  The  letters  may  be  numbered  forward,  back- 
ward, or  arbitrarily. 

Illustration.     Key:  Now  be  quick. 

(1)  NOWBEQUICKor  (2)  NOWBEQUICK 
123     4567890  098    7654321 

By  using  different  keys,  the  cost  may  be  concealed  from  all 
except  marker  and  manager,  and  the  selling  price  from  all 
except  marker,  manager  and  selling  force.  There  is  a  tend- 
ency, however,  to  mark  the  selling  price  in  plain  figures. 

In  writing  with  a  key,  the  last  two  letters  represent  cents. 
If  any  figure  occurs  twice  in  succession,  an  arbitrary  letter, 
called  a  repeater,  is  used  the  second  time.  Arbitrary  letters 
are  sometimes  used  for  fractions,  but  are  seldom  needed,  as 
fractions  of  a  cent  are  uncommon  in  the  marking  of  such  goods 
as  are  commonly  "tagged." 

Illustration.  Example.  Write  the  marking  tag  for  canes,  no.  221, 
costing  $2.80  and  selling  for  $3.60,  using  the  key  (1),  as  above. 

Solution.  Substituting  the  corresponding  letters  for  the  figures,  $3.60 
is  written  "wqk,"  and  $2.80  "oik."     The  tag  is  written 

Note.     If  the  selling  price  were  $4.tX),  and  s  were 
repeater,  the  price  would  be  written  "bks." 


No.  221 
wqk 
oik 


EXERCISE. 

1.  Interpret  the  following,  based  on  the  key  "hypodermic " : 

peo^      hoc^      iy^      poxc^      opd^      hmr,      "x"  is  repeater, 
yic        hex      er       ydcx       pyc        rm 

2.  Write,  using  the  key  "Mayflower,"  and  any  repeater: 

37      4m      675      19.20      1.10      99      4J5      12.32 
5'     3.80'     5.40'     11.85'      65'     80'     2.60'      9.64* 

3.  Repeat  (2),  writmg  sellmg  price  in  figures,  and  the  cost  in  the  key 
Now  be  quick." 


234 


BUSINESS   ARITHMETIC. 


Write  the  cost  and  selling  price  in  the  following  examples,  using  these 
keys: 

Cost  Key. 

-       I       \ 

/      T       +       ±       =F       1 

n     O 

1         2        3 

4        5        6        7        8        9 
SelUng  Key. 
WASHING  TUB  M 

0       R 

' 

1234567890R 

First  Cost. 

Charges. 

Gain.        x 

1.    $2.40 

$  .30 

20% 

2.      3.25 

.45 

25%       • 

3.        .88 

m% 

15% 

4.     14.56 

.60 

30% 

First  Cost. 

Sell.  Ex. 

Gain  or  Loss. 

5.     $2.40 

5%  of  S.  P. 

.    20%    ofS.  P. 

6.      3.82 

10%  of  S.  P. 

25%    ofS.  P. 

7.      9.60 

10%  of  S.  P. 

20% 

8.     15.00 

$1.00 

25% 

Using  the  key  "M 

ayflowers,"  write  cost  and  selling 

price  per  article 

for  the  following: 

Cost.        Per. 

Gain.                   Cost.       Per. 

Expense.     Gain. 

9.     $9.00        dz. 

25%.           12.    $15.00    dz. 

$.72           20% 

10.      4.80        C. 

20%            13.       7.20      gr. 

5.00           25% 

11.      6.40  cas.  (2  dz.)  15%.            14.       1.26      1/2  dz. 

.12c         20% 

15.    Using  the  key  "Washing  tub,"  mark  the  goods  in  the  following 
invoice  at  an  advance  of  25%  on  gross  cost. 


Messrs.  Topham  and  Chase, 

Pittsburgh,  Pa. 
Bought  of  C.  R.  Newman  &  Co. 
Terms :  60  days. 


Philadelphia,  January  29,  19 — . 


1829 

47 

639 

426 

358 


2i 

U 

3 

4 

3 


dz.  Neckwear, 


Outing  Shirts 
Caps, 
Less  10%. 


3.60 
2.80 
4.50 
9.60 
3.25 


Expressage,  $2.80. 
Selling  Expense,  10%. 


ELEMENTARY  PROFIT  AND   LOSS.  235 

Individual  Original  Work. 

Design  a  marking  sheet  showing  first  cost  of  each  class  of  goods  in 
bulk,  cost  per  article,  discounts,  gains,  terms,  selling  and  list  prices,  etc. 
Make  entries  for  at  least  fifteen  articles. 


CHAPTER  XXXI.     . 

COMMERCIAL  DISCOUNT. 

One  of  the  most  comm9n  and  important  business  appli- 
cations of  percentage  is  the  expression  and  reckoning  of  trade 
discounts. 

ILLUSTRATIVE    EXERCISE. 

1.  On  Jan.  6,  J.  P.  Robertson  buys  60  bbl.  flour  @  $5.00.  He  agrees 
to  pay  in  90  days,  but  is  offered  3%  off  (discount  from)  the  debt  if  he  pays 
within  ten  days.  Up  to  Jan.  16,  $  ?  will  settle  the  bill;  after  that,  and  until 
April'O,  $?. 

Note.  These  terms  may  be  written  "90  days  net,  3%  10  da."  or  "Net 
/90;  3/10."  In  business,  the  credit  terms  are  usually  30  da.,  60  da.,  or 
90  da.  Rates  of  discount  vary  with  firm  and  customer.  Interest  is 
sometimes  charged  for  overdue  payments. 

2.  A  Chicago  mail  order  house  allows  2%  discount  on  orders  of  $10; 
3%  on  orders  of  $20;  5%  on  $50  orders,  etc.  What  discount  should  a 
customer  receive  on  a  $20  order?     (Why?) 

3.  The  Virginia  Brick  Co.  allows  5%  discount  on  orders  for  over  10  M. 
brick.     26  M.  @  $12  per  M.  cost  what  sum? 

4.  The  catalog  price  of  5"  metal  piping  is  $1.40  per  foot.  The  market 
price  is  20%  less.     1000  ft.  at  the  market  price  cost  what  sum? 

Note.  Market  prices  of  some  articles  vary  with  demand  and  with  the 
cost  of  raw  materials  used  in  their  manufacture.  To  avoid  constant  re- 
printing of  trade  catalogs,  prices  are  listed  high,  and  customers  are  notified 
of  current  discounts  on  goods  that  they  desire. 

5.  The  Sandford  Pub.  Co.  issues  a  nature  book  to  retail  at  $1.80. 
The  book  is  sold  to  dealers  at  20%  discount,  enabling  them  to  realize  what 
per  cent,  of  gain? 

Note.  Publishers  and  some  manufacturers  fix  the  retail  price  of  their 
products,  selling  to  dealers  and  jobbers  at  a  discount,  and  to  private 
customers  at  'list.' 

Solution.     $1.80  Retail  price. 

20%  or  1/5  of  $1.80  =  $.36,  discount. 
$1.80  -  $  .36  =  1.44,  cost  to  dealer. 

$  .36  -^  $1.44  =    .25  or  1/4,  or  25%,  the  rate  of  gain. 

236 


COMMERCIAL  DISCOUNT.  237 

It  is  evident  that  a  commercial  discount  is  an  allowance  or 
subtraction  from  the  list  or  marked  price  of  goods  for  such 
business  reasons  as:  (1)  payment  before  due;  (2)  size  of  order; 
(3)  reduction  of  list  to  market  price;  (4)  to  meet  competition; 
(5)  to  allow  for  profits  to  middlemen  who  are  expected  to 
sell  'at  list,'  etc.  These  discounts  commonly  are  expressed 
as  rates  per  cent.  Many  discounts  are  aliquot  rates,  thus 
permitting  short  methods  of  computation. 

EXERCISE. 
(Answer  orally  if  possible.) 

1.  To  what  terms  in  percentage  do  the  following  correspond:  Rate, 
of  discount,  list  price,  discount,  reduced  or  net  price? 

2.  Find  the  per  cent,  of  list  price  paid  for  goods  bought  at  one  of  the 
following  discounts:  40%  (suggestion,  100%  -  40%),  30%,  28^%,  63%, 
75%,   1/3%. 

3.  Determine  the  net  cost  of: 

200  ft.  of  piping  @  $1.20,  less  20%. 
200  pr.  gloves  @  $3.60,  less  12^%. 
A  $500  piano,  less  40%. 
40  valves  @  $.75,  less  15%. 

4.  How  do  terms  of  "90  days,  4/30"  affect  the  settlement  of  a  $950 
bill  of  goods,  bought  February  7  and  settled  March  1? 

5.  Check  up  the  Randall  Mfg.  Co.'s  bill  of  $552,  as  follows:  6  sewing 
machines  @  $40,  less  25%,  $180;  4  washing  machines,  @  $27,  less  30%, 
$72;  3  farm  wagons,  @  $85,  less  20%,  $202. 

EXERCISE. 

A  bilhng  clerk  handled  and  checked  the  following  transactions  on 
Oct.  24.     Find  missing  values. 

1.  Received  of  S.  P.  Adams,  check  for  $ in  full  settlement  for 

1200  lb.  Japan  tea,   @  23c,  bought  on  Oct.  18,  on  terms  of  60  da.  net; 
3/10. 

2.  Mailed  Waukeska  Canning  Co.  check  for  $ ,  for  invoice  of 

Oct.  22,  300  bbl.  Cerota  Flour,  1/4  s.  (i.  e.,  1/4  bbl.)   @  $5.80  per  bbl. 
Terms,  30  da.;  1%,  10  da. 

3.  Checked  Parker  Bros,  invoice  of  120  bbl.  S.  T.  Flour  @  $4.10; 
12  sacks  Graham  Flour,  @  $2.80;  40  doz.  C.  Tomatoes  @  $.92;  144  bx.  K. 


238 


BUSINESS   ARITHMETIC. 


Soap,  @  $7.00.  Discount,  12^%.  Terms:  30  day  note,  $400;  cash  for 
balance.     Mailed  check  for  $ . 

4.     Received  check  for  $ from  Franklin  Cereal  Co.,  being  rebates 

on  September  sales  of  $746.20,  at  12^%. 

Note.  Manufacturers  a  certain  lines  of  groceries  sell  to  wholesalers 
at  prices  at  which  the  latter  sell  to  retailers.  At  intervals  the  manu- 
facturers return  wholesalers  a  discount  or  rebate,  based  on  the  amount 
of  their  monthly  sales. 

Series  of  Discounts.  Often  two  or  more  discounts  are 
allowed  on  the  same  item;  one,  for  example,  for  quantity, 
and  a  second  for  early  payment.  In  such  cases,  one  discount 
is  reckoned  on  the  list  price;  the  second  on  the  list  price  minus 
the  first  discount  and  so  on. 

Illustration.  Successive  discounts  of  33^%,  20%,  5%,  are  allowed 
on  a  shipment  of  4000  ft.  iron  pipe  @  $1.50,  reducing  the  cost  to  $ . 


Solution  (a). 
4000  X  $1.50 
331%  =  1/3 

20%  =  1/5 

5%  =  1/20 

Solution  {b). 


1/3  of  6000 
1/5  of  4000 
1/20  of  3200 

100% 


=  $6000  list  price. 
=  2000  first  discount. 

$4000  second  price. 
=     800  second  discount. 

$3200  third  price. 
=     160  third  discount. 

$3040  net  cost. 


5%  of  80% 
33i%of76%  =  iof76 


=list  price. 
100%,  list  price. 
20      one  discount. 
80%  second  price  in  per  cent. 
=     4%  second  discount. 

76% 
=_25i%  third  discount. 
50f  %  net  cost  in  per  cent. 
501%  of  $6000  =  $3040. 
Note.     100%  —  50|%  =  49|%,  which  is  called  the  single  discount 
equal  to  the  series. 

FOR    PROOF    OR    DISCUSSION. 

1.  Why  does  it  pay  a  dealer  to  allow  discounts  for  the  causes  men- 
tioned? 

2.  Proposition.    The  order  in  which  the  discount  rates  for  a  series 
3xe  used  does  not  affect  the  final  result. 


COMMERCIAL  DISCOUNT.  239 

3.  Illustrate  with  the  series  33^%,  40%,  10%,  the  possible  arith- 
metical advantage  of  changing  the  order  of  discounts. 

4.  Proposition.  The  single  discount  equal  to  a  series  is  always  less 
than  its  sum. 

5.  Proposition.  The  single  discount  equivalent  to  a  series  of  two 
discounts,  equals  the  difference  between  the  sum  and  product  of  the  series. 

ORAL    EXERCISE. 

1.  Find  the  cost,  in  per  cent,  of  list  price,  if  goods  are  bought  at  discoimt 
of:  (See  solution  (6),  illustration  2,  page  238.) 

40%,  10%,  5%.  5%,  10%. 

80%,  3%.  50%,  20%,  10%. 

161%,  10%,  25%.  10%,  10%,  10%. 

10%,  33i%,  40%.  3  20s  (i:  e.,  20%,  20%,  20%.) 

2.  Find  the  single  discoxmts  equivalent  to  the  following  series: 

40%,  25%.        20%,  25%.     10%,  25%,  33^%.     16|%,  20%. 
50%,  10%.        50%,  50%.     10%,  15%  20%,  25%,  5%. 

3.  Required,  the  net  cost: 

(a)  Of  400  lb.  castings  @  10c,  less  20%,  25%. 
(6)  Of  50  pr.  gloves  @  $2.00,  less  10%,  25%. 

4.  Compare  these  offers  for  goods  of  the  same  quality: 

(a)  List  price  $800.     Discounts  25%,  33i%,  10%. 
(6)  List  price  $600.     Discounts  16|%,  25%. 

5.  What  case  of  percentage  is  involved  in  these  examples. 

EXERCISE. 

1.  Find  the  net  receipts  from  the  following  sales  of  merchandise: 

$326.50  at  discounts  of  10%,  20%. 
$415.10  at  discounts  of  30%,  5%. 
$300.00  at  discounts  of  40%,  20%,  10%. 

2.  Bought,  Aug.  30,  on  terms  90  da.,  3/30,  5/10,  4  dz.  knives  @  $8.00, 
less  15%;  2  dz.  sauce-pans  @  $3.40,  less  10%,  10%;  3i  dz.  wash  boilers 

@  $38.80,  less  30%,  5%.     Paid,  Sept.  16,  $ . 

3.  A  library  has  an  opportunity  to  purchase  books  at  40%,  5%  off. 
A  purchase  fund  of  $420  per  year  will  buy  $ worth  of  books  at  list  prices. 

4.  In  selling  a  bill  of  80  rockers  @  $8.40,  less  25%,  10%,  a  clerk  reckons 
the  equivalent  discount  at  31%.  What  overcharge  may  the  customer 
claim? 

5.  Both  BrowTi  &  Co.  and  A.  C.  Lewis  offer  me  8000  ft.  of  choice  cypress 
@  $80  per  M.,  but  Brown  &  Co.  quote  discqimts  of  33^%,  5%;  while 


240  BUSINESS   ARITHMETIC. 

Lewis  quotes  20%  and  20%.     WTiich  is  the  better  ofifer?     By  accepting  it, 
$ is  saved. 

Rates  of  discount.     In  wholesaling  and  manufacturing,  there 

is  constant  necessity  to  calculate  rates  of  discount — owing  to 

fluctuations  in  market  prices  and  other  causes. 

Illustration.     An  article  listed  at  $9.00  drops  to  a  market  value  of 
$4.00.      33i%   discount    has   already    been   quoted    customers.      What 
further  discount  rate  must  be  allowed? 
Solution. 

The  list  price  =$9.00 

33i%  discount  is  1/3  of  $9.00        =  3.00 

First  discount  price  6.00 

Required  price  4.00 

Additional  discount  to  be  allowed  2-00 

$2.00  dis.  must  be  reckoned  on  $6.     It  =  1/3  of  $6,  or  33|%.     Therefore 

the  second  discount  is  also  33 i%. 

ORAL    EXERCISE. 

1.  Discounts  of  20%  and  ?  %  yield  a  selling  price  of  60%  of  list. 

2.  Discounts  of  ?  %  and  ?  %  enable  one  to  sell  at  50%  of  list. 

3.  Discounts  of  $20.25%  and  ?  %  lower  a  list  price  of  $380  to  a  selling 
price  of  $180. 

EXERCISE. 

1.  What  discount  in  addition  to  one  of  15%  will  lower  a  list  price  of 
$7.20  to  a  market  price  of  $5.00? 

2.  The  Essex  chair  manufactured  at  a  cost  of  $5.00  retails  at  $8.40. 
The  manufacturer  lists  at  retail  price.  If  he  is  to  gain  20%,  he  can  allow 
the  retailer  what  rate  of  discount? 

Suggestion.     Find  the  manufacturer's  selling  price.     Compare  with  list. 

3.  The  same  manufacturer  Hsts  at  $10  an  article  costing  $6.40  to  make. 
If  dealers  sell  at  list  price,  they  should  buy  at  what  discount  in  order  that 
they  and  the  manufacturer  may  make  an  equal  money  profit? 

EXERCISE. 
(Rates  of  profit  and  loss.) 
Find  missing  values. 

Questions  of  profits  or  losses  on  individual  items  sometimes  arise. 
Thus  a  dealer  may  wish  to  determine  his  rate  of  profit  on  carpet  bought 
at  $2.00  and  sold  at  $4.00  less  20%,  30%.     By  comparing  net  selling  price 


COMMERCIAL   DISCOUNT.  241 

with  cost,  he  finds  the  rate  to  be %.     Or  he  discovers  that  a  clerk  has 

lost  him  $ by  discounting  at  45%  a  $120  sale  on  a  series  of  20%  and 

25%. 

Sometimes  the  difference  of  measure  affects  the  problem,  as  in  com- 
puting the  rate  of  profit  on  step-ladders  bought  @  $12.00  per  dz.  less  50%, 
5%  and  selling  at  $1.20,  findmg  it  to  be %. 

Perhaps  a  case  arises  in  which  part  of  an  invoice  has  been  sold.  Say, 
20  out  of  35  bookcases,  bought  @  $8.00  less  40%,  10%  have  been  sold 
at  list,  and  the  balance  at  20%  profit.     This,  he  finds,  yields  him  a  profit  of 

%. 

Questions  of  comparative  sales  arise.  Goods  costing  $6.00  may  be 
fisted  at  $16.00.  An  increase  of  discount  rate  from  40%  to  50%  results 
in  increasing  average  sales  30%.     "Does  it  pay?"  he  asks,  and  finds  that 

.     Or,  a  $20.00  chair,  costing  $8.40  to  make  is  usually  discounted  30% 

to  the  trade.  By  experiment  over  a  long  period,  the  dealer  finds  that  his 
average  sales  per  week  increased  from  40  to  50  when  he  allowed  10%  ad- 
ditional discount.     By  percentage  he  finds  that  his  present  rate  of  profit 

is %,  where  formerly  it  was %. 

Some  cases  of  marking  goods  arise. 

Illustration.     Mark  a  desk,  costing  $15.00  to  manufacture,  at  a  price 
to  yield  the  maker  33  §%  profit,  after  allowing  discounts  of  40%   and 
16f  %  to  the  trade. 
Solution. 

Let  100%  =fist  price. 

100%  -  40%  =60%,  1st  discount  price. 

161%,  or  1/6  of  60%    =10% 

60%  —  10%  =?50%,    second    discount    price,    or    selling 

price  per  cent. 
33i%  of  $15.00  =$  5.00  profit  to  maker. 

$15.00 +$5.00  =$20.00  selling  price. 

.-.  50%  of  list  =$20.00 

100%  =$40.00,  fist  price. 

EXERCISE. 

1.  $8.00  rockers,  bought  at  discounts  of  25%  and  5%,  must  be  sold 
at  $ to  gain  20%. 

2.  At  what  price  must  articles  costing  $18.00  be  marked  to  allow  for 
331%  profit  and  5%  bad  debts? 

3.  A  manufacturer's  patent  extension  table  cost  $8.40  to  manufacture. 
If  he  is  to  allow  for  20%  profit  for  himself,  and  for  20%  discount  to  the 
wholesaler,  he  must  name  what  list  price? 

17 


242 


BUSINESS   ARITHMETIC. 


4.  The  public  does  not  like  to  pay  more  than  $7.20  for  a  certain  article. 
Allowing  for  a  manufacturer's  profit  of  20%,  and  a  dealer's  discount  of 
20%,-  the  maker  must  strive  to  keep  the  cost  of  manufacture  within  $ — 

EXERCISE. 

1.  Sold  to  Robt.  M.  Drake,  Morristown,  N.  J.,  via  DLW  frt.  30  days 
2/10;  40  doz.  files  @  $2.60,  less  40%,  10%;  3i  doz.  hand  saws,  No.  82,  @ 
$17.50,  less  25%,  10%.    Write  bUl. 

2.  Compare  bids  received  for  the  purchase  of  100  C  machine  bolts 
3/8"  X  6  1/2,  needed  for  stock. 

R.  C.  Shafer  offers  them  @  $4.00  less  50%,  10%,  5%. 

James  Casler  offers  them  @  $3.90  less  60%,  10, 

Newark  Bolt  Co.  offers  them  @  $4.10  less  60%,  15%,  5%. 

3.  Extend  this  bill. 

Pittsburgh,  Pa., ,  191— 

The  Morris  Hardware  Co. 
New  York  City. 
Bought  of  The  Randall  Iron  Co. 
Terms:  60  days;  3/10 


2 

4^ 


2i 


C  Machine  bolts,  1/2X5 
C  Machine  bolts  7.20    7.52 
5/8X4"     4^' 
50%,  15%,  5% 


dz.  pr.  Hinges,  No.  382,  8' 
Less  60%,  10% 

dz.  Steel  Squares  No.  11  (< 
Less  30%,  10%, 


5.60 

-I 



— 

4.20 

2. 

— 

— 

*  Note.     4^  hundred  machme  bolts  5/8"  X 4"  @  $7.20  per  100,  and  the 
same  quantity  5/8"  X4^"  @  $7.52. 


EXERCISE. 

(Give  an  illustration  in  each  case.) 

1.  What  values  must  be  known  to  determine:  (a)  list  price,  (6)  missing 
discount  rates,  (c)  net  cost,  (d)  discount  in  money,  (e)  rate  of  profit  or  loss? 

2.  What  values  can  be  determined  from:  (a)  list  price  and  series  of 
discounts;  (h)  series  of  discounts,  two  as  rate,  one  also  as  money;  (c)  cost 
and  series;  (d)  cost,  desired  profit,  and  rate  of  discount;  (e)  two  put  of 
three  rates,  cost  and  list  price? 


COMMERCIAL   DISCOUNT.  243 

It  is  not  at  all  necessary,  in  the  study  of  many  questions  of 
profits,  discounts,  and  even  of  questions  of  business  policy 
involving  direct  financial  results,  that  the  fundamental  calcu- 
lation, or  the  calculations  to  determine  an  advisable  course, 
should  be  based  on  money  values.  Percentage  is  a  valuable 
means  of  studying  business  actions,  somewhat  abstractly,  and 
of  deducing  results  as  rates  per  cent.,  that  are  practically 
formulas. 

Illustration.  A  question  like  this  may  arise:  If  merchandise  is 
bought  at  40%  discount  from  list,  and  sold  at  list,  how  much  of  the  selling 
price  may  be  allowed  for  discounts  and  selling  expense,  and  yet  realize 
25%  profit.  The  cost  equals  60%  of  Hst.  25%  profit  on  cost  equals 
15%  of  list.  The  desired  selling  price  equals  75%  of  list.  100%  list —75% 
=  25%.     Therefore  selling  expense  and  discounts  must  be  kept  within  25%. 

EXERCISE. 

Each  example  in  this  exercise  should  be  solved  without  assuming  any 
money  value — although  the  resulting  answer  may  be  applied  as  a  rate 
per  cent,  to  an  infinite  number  of  prices  and  money  values.  Find  missing 
values. 

(a)  Questions  Primarily  of  Discounts. 

1.  The  portion  of  the  list  price  paid  for  goods  bought  at  34 1%  discount 
is  -%  . 

Note.  The  answer,  65  |%,  is  general  for  the  given  discount,  regard- 
less of  price  concerned. 

Apply  this  answer  to  finding  the  cost  of  articles  listed  respectively  at 
$4.20,  $5.00,  20c,  $340. 

2.  How  do  terms  of  "90  da.,  3/60,  5/10"  affect  the  settlement  of  any 
bill  paid  within  five  days  of  date  of  purchase? 

3.    %  of  the  list  price  is  paid  for  goods  bought  at  discounts  of 

40%  and  10%.    Apply,  to  finding  the  cost  of  goods  listed  at  $486,  and  at 
$56. 

4.  The  difference  between  a  series  of  10%  and   40%,  and  a  single 

discount  equal  to  their  sum  is %.    What  is  the  money  difference  on  a 

list  price  of  $840? 

5.  Which  is  the  better  *f or  a  seller  to  offer — any  series  of  discounts  or 
its  sum? 

6.  Discounts  of  25%  and  x%  will  lower  a  list  price  50%. 


244  BUSINESS   ARITHMETIC. 

7.  Determine  a  series  of  two  discounts  to  lower  a  list  price  45%. 
Suggestion.     Assume  any  rate  less  than  the  total  discount,  for  example, 

20%.     Calculate  the  other  rate.     How  many  solutions  are  possible? 

8.  The  separate  discounts  in  a  series  of  3  20s  compare  how  in  real 
value?  Illustrate,  after  obtaining  the  proportional  value,  by  application 
to  a  list  price  of  $8000. 

9.  Determine  a  series  of  two  discounts,  equivalent  to  a  single  discount 
of  40%,  which  shall  be  equal  in  numerical  value. 

Suggestion.  Divide  the  single  discount  into  equal  parts,  and  find  the 
equivalent  rates.  The  principle  applied  in  this,  and  in  one  or  two  of  the 
following  examples,  is  applied  by  producers  in  allowing  for  middlemen's 
profits. 

10.  Determine  a  series  of  three  discounts  equivalent  to  a  single  discount 
of  48%,  and  equal  in  numerical  value. 

11.  A  series  of  two  discounts  is  equal  to  45%  off  list,  but  the  first  dis- 
count is  twice  the  second  in  real  value.     The  series  is  —  %  and  —  %. 

(6)  Questions  Primarily  of  Profit  and  Loss. 

12.    %  is  reaUzed   by  buying  at  10%  and  33  i%  discount  from 

the  fist  price  at  which  the  goods  are  sold? 

13.  When  a  publisher  sells  his  net  books  to  dealers  at  40%  discount, 
the  latter  reahze  ?  %  profit. 

14.  Is  it  better  for  a  retailer  to  sell  at  30%  advance  on  cost,  or  to  sell 
at  a  list  price  from  which  his  wholesaler  has  allowed  him  30%  discount? 

15.  Compare  these  offers,  based  on  the  same  Ust  price:  (a)  Discount 
20%,    5%,  10%.     (6)  Dis.  of  3  20s.     (c)  Dis.  of  30%,  10%,  15%. 

16.  My  stock  is  bought  at  40%  discount  from  the  list  price  at  which 
I  sell,  but  the  expenses  of  sale  average  15%  of  the  selling  price.  What 
net  rate  of  profit  is  realized? 

17.  A  50%  increase  in  sales,  resulting  from  allowing  20%  discount, 
has  what  effect  on  former  profits  of  30%  made  by  selling  at  list? 

18.  Merchandise  is  bought  at  40%  discount  and  sold  at  20%  discount 
from  the  same  list  price.     What  is  the  effect  of  interchanging  rates? 


CHAPTER  XXXII. 

AGENCY. 

An  agent  is  one  who  does  business  for  another.     The  person 

whom  he  represents  is  termed  the  principal. 

Illustrations.  (1)  Jones,  at  your  request,  buys  you  a  saddle  horse. 
(2)  Jefferson  buys  for  Chase  400  shares  P.  R.  R.  stock.  (3)  A  farmer  ships 
a  carload  of  potatoes  to  a  city  produce  dealer  who  returns  the  farmer  the 
money  proceeds  from  their  sale,  less  his  charge  for  selling. 

EXERCISE. 

1.  Give  other  illustrations  of  agency  in  everyday  affairs. 

2.  Name  the  agent  and  the  principal  in  each  illustration. 

3.  Why  are  agents  required  in  business? 

4.  State  some  of  the  advantages  and  disadvantages  of  dealing  through 
agents. 

5.  Name  common  businesses  that  are  agency  businesses. 

In  certain  businesses  the  agent  is  called  a  broker  (stocks, 
bonds,  notes,  etc.),  or  a  factor  (cotton),  or  a  commission 
merchant  (produce).  Drummers,  buyers,  attorneys,  auction- 
eers and  salesmen  are  agents.  The  agent  may  act  without 
pay,  may  receive  a  salary,  or  may  receive  a  payment,  often 
termed  a  commission,  based  on  the  quantity  or  value  of  goods 
bought  or  sold.  He  may  receive  both  salary  and  commission. 
The  pay  of  a  broker  is  termed  brokerage  and  that  of  a  factor, 
commission.  The  terms  fee,  share  and  allowance  are  also  com- 
mon. Per  cent,  commissions  are  based  on  the  exact  amount 
paid  (net  cost),  or  received  (gross  proceeds)  by  the  agent  for 
the  goods — regardless  of  charges. 

The  principal  pays  his  agent  for  a  'purchase,  the  net  cost 
plus  all    charges  for  freight,  drayage,  commission,  etc.  (the 

245 


246  BUSINESS  ARITHMETIC. 

gross  cost).  He  receives /rom  a  sale  the  gross  proceeds  minus  all 
charges  (the  net  proceeds). 

Guaranty  is  an  additional  fee  paid  the  agent  for  taking  the 
risk  of  securing  payments  from  customers  to  whom  he  sells. 
Warranty  is  an  additional  fee  paid  the  agent  for  ensuring  the 
quality  of  goods.  The  agent  pays,  but  charges  to  his  principal, 
freight,  drayage,  storage,  insurance,  inspection  charges,  etc. 

The  agent  acts  for  his  principal.  The  principal,  as  a  con- 
signor, consigns  or  ships  goods  to  the  agent,  or  consignee,  to 
be  sold.  The  principal  calls  the  goods  a  shipment;  the  agent 
calls  them  a  consignment. 

EXERCISE. 

1.  After  carefully  reading  the  preceding  sections,  define  agent,  broker, 
commission,  guaranty,  warranty,  principal,  gross  proceeds,  gross  cost, 
the  sum  earned  by  the  agent,  the  sum  the  principal  receives  from  a 
sale. 

2.  Compare  the  following  with  the  terms  of  percentage:  agent's 
commission,  rate  of  commission,  gross  proceeds,  net  proceeds  (if  com- 
mission is  only  charge). 

1.  PAYMENT  OF  AGENTS. 

The  rate  of  commission  is  expressed  as  a  lump  sum,  as  a  rate 
per  cent,  based  on  the  price  at  which  the  agent  buys  or  sells, 
as  a  fractional  part  of  this  value,  or  as  a  rate  per  article. 

Illustrations.  I  pay  Scott  $25  for  selling  my  horse  for  me.  My 
agent  sells  200  bbl.  of  my  potatoes  at  $2,  and  charges  me  5%  of  the  selUng 
price  for  his  services.  I  offer  one-fifth  of  the  sum  he  obtains  to  a  collector 
who  secures  payment  of  a  debt  due  me. 

Example.  What  does  an  agent  earn  for  buying  or  selling  200  bbl.  of 
flour  @  $4.80,  on  5%  commission? 

Solution.     The  selling  price  =  200  X  $4.80  =  $960. 
The  commission  =  5%  of  $960,  or  $48. 

It  is  evident  that  the  agent's  per  cent,  of  commission  is 
reckoned  on  the  price  he  pays  or  receives  for  the  goods. 


AGENCY.  247 

ORAL   EXERCISE. 

Find  the  commission: 

1.  On  400  bbl.  @  2c  per  bbl. 

2.  On  2000  bu.  @  l/4c. 

3.  On  a  $3600  purchase  at  2|%  commission;  at  4%;  at  5%;  at  25%= 

4.  On  1200  bbl.  at  15c  per  bbl. 

5.  On  800  bbl.  purchased  @  $2  on  5%  commission. 
What  is  the  agent  paid  for  his  services? 

6.  For  the  collection  of  a  $1200  debt  on  5%  commission? 

7.  For  auctioning  off  $1800  worth  of  goods  on  3%  commission? 

EXERCISE. 

1.  Compare  a  5%  rate  of  commission  with  a  rate  of  l/2c  per  bu.  on 
merchandise  costing  30c  per  bu. 

Suggestion.    Reduce  the  second  rate  to  a  rate  per  cent. 

2.  Compare  a  rate  of  3%  with  a  rate  of  15c  per  bx.  on  goods  selling 
at  $2.40  per  bx  Is  it  possible  to  make  a  comparison  in  the  above  case 
if  no  money  value  is  given? 

3.  C.  Johnson. 


$  40.00  $102.10 

$400.00 
$206.40 
An  agent  succeeds  in  collecting  three-fourths  of  this  debt  for  me.     If 
his  commission  is  6%,  he  earns  $  — . 

2.  SALES. 
INTRODUCTORY    EXERCISE. 

1.  You  sell  for  me  10  T.  timothy  on  5%  commission.  The  customer 
pays  you  what  sum?     You  charge  $  ?,  sending  me  what  sum? 

Note.  The  principal  receives  the  selling  price  (gross  proceeds)  — 
charges,  or  the  net  proceeds. 

2.  An  agent  secures  a  purchaser  for  a  house  and  lot  who  pays  $12,600. 
The  agent  pays  $30  for  survey,  $10  for  transfer,  and  charges  5%  commission. 
What  does  the  agent  receive  and  what  sum  does  he  pay  to  other  parties? 

3.  Whose  money  does  the  agent  use  to  pay  for  charges?    Explain. 
The  standpoint  of  each  party  to  an  agency  transaction  is 

different,  and  the  calculations  in  which  he  is  directly  inter- 


248  BUSINESS   ARITHMETIC. 

ested  vary.     Recognition  of  "viewpoint"  is  often  essential  to 
accurate  analysis  and  solution  of  agency  problems. 

Illustration.  A.  B.  Newton  ships  to  Robert  Kent,  commission 
merchant,  200  bbl.  apples  which  the  latter  sells  at  $2.40.  He  pays  freight 
of  15c  per  bbl.,  and  charges  5%  commission.  He  mails  to  Newton  a 
check  for  $  ?,  net  proceeds. 

From  the  standpoint  of  Newton  the  problem  is:  "I  ship  Robert  Kent 
200  bbl.  apples  which  he  sells  at  $2.40,  on  5%  commission,  paying  15c 
freight  per  bbl.     I  should  receive  $  ?  " 

From  the  standpoint  of  the  freight  agent:  "What  is  the  freight  on  200 
bbl.  at  15c?" 

From  the  standpoint  of  the  agent  (To  customer) :  "What  must  I  receive 
for  200  bbl.  at  $2.40?"  (To  freight  agent):  "What  freight  is  due  on  200 
bbl.  at  15c?"  (For  himself) :  "What  is  my  commission  on  this  sale  at  5%?" 
(To  principal) :  "  Considering  these  previous  transactions,  what  do  I  owe 
Newton?" 

From  the  standpoint  of  the  customer:  "Two  hundred  bbl.  apples  at 
$2.40  will  cost  me$?" 

Solution.     200  X $2. 40  =  $480  gross  proceeds,  or  sum  paid  by  customer. 
200  X  15c     =  $  30  freight  paid  to  freight  agent  by  agent. 
6%  of  $480  =  $  24  commission. 
$  54  total  charges. 
Gross  proceeds  — charges  =  net  proceeds,  due  principal. 
$480— $54  =  $426,  net  proceeds. 

ORAL   EXERCISE. 

Memoranda  of  Sales. 

(a)  400  bu.  oats,  @  40c,  commission  l/4c  per  bu. 

(6)  5  copies  Marston's  Business  Encyclopedia,  $14,  on  5%  commission. 

(c)  A  horse,  $240;  wagon  $90;  and  harness  $30.  Commission  3%; 
misc.  expense  $4. 

(d)  20  crates  eggs,  30  doz.  each,  30c  per  doz.  Commission  5%;  express, 
50c  per  crate. 

1.  What  do  agent  and  principal  receive  in  each  case? 

2.  Any  increase  in  general  charges  affects  what  parties?    Illustrate. 

EXERCISE. 

Before  solving,  re-word  the  transaction  from  each  party's  standpoint, 
and  state  necessary  and  unnecessary  values  for  each  computation. 

1.     The  consignor  should  receive  a  check  for  $ ,  as  proceeds  of  20 

crates  eggs,  30  doz.  each,  @  18c.    Commission  6%. 


AGENCY. 


249 


2.  Find  the  sum  due  the  principal  on  a  shipment  of  60  T.  timothy 
hay  @  90c  per  C.     What  sum  does  the  agent  earn  at  4%  commission? 

3.  S.  P.  Naylor  sells  for  James  Carter  12,000  bu.  com  at  50  l/4c, 
commission  l/4c  per  bu.     What  sum  does  Naylor  receive? 

Note.  In  the  following,  name  the  new  factors,  and  discuss  their  effect, 
and  find  missing  values. 

4.  I  shipped  M.  C.  Parker  360  bbl.  apples  which  he  sells  at  the  market 
quotation  of  $2.90;  on  4%  commission  and  1%  warranty. 

The  customer  pays  $ ;  Parker  receives  $ ;  he  earns  $ ;  and 

he  pays  me  $ . 

5.  J.  C.  Jones  receives  from  his  agent  a  check  for  $60.40  as  net  proceeds 
from  a  sale  of  three  hundred  pounds  of  butter  at  27c,  commission  53^%, 
miscellaneous  charges  $4.75.     Check  up  the  returns.     The  customer  paid 


6.  250  bbl.  apples,  shipped  to  be  sold  at  the  market  quotation  of  $3.20, 
arrive  on  a  falling  market  and  are  sold  at  $2.90,  drayage,  10c  per  bbl., 
commission  5%.     What  effect  has  the  fall  in  price  on  the  net  proceeds? 

7.  The  consignor  should  receive  $- net  proceeds  from  a  shipment 

of  16,000  bu.  oats,  disposed  of  at  41c.     Freight,  2c  per  bu.;  insurance  l/2c; 
storage,  l/2c;  commission,  l/4c. 

8.  Extend 

Account  Sales. 
From  John  C.  Carter, 

Baltimore,  Md.,  Jan.  17,  1913. 
To  Robert  G.  Mason, 

Hyattsville,  Maryland. 
For  Merchandise  Received   Jan.  2,  1913,  via     B.  &  O. 
Consigment  No.  4726  FoHo  88  R.  M.  Green       Clerk. 


Jan. 


Sales 


100  lb.  Dressed  chicken 
450  lb.  Dressed  turkey 
620  lb.  Dressed  chicken 


.16 
.18 
.17 


Charges. 


Expressage  $2.45  Drayage  $1.10  Storage  $3.15 
Inspection  $  Insurance  $  Commission  $  (4^%) 
Other  Charges  $  Total  Charges 

Net  Proceeds 


250  BUSINESS  ARITHMETIC. 

3.    PURCHASES. 
INTRODUCTORY    EXERCISE. 

1.  You  buy  through  an  agent,  on  l/8c  per  bu.  commission,  6000  bu. 

wheat  at  90c.     The  agent  pays  $ —  for  the  grain,  charges  $ commission, 

and  you  pay  $ .    What  effect  have  charges  on  the  cost  to  the  prin. 

cipal? 

2.  An  agent  buys  for  his  principal  80  bbl.  winesap  apples  @  $2.50  on 
5%  com.,  and  50c  per  bbl.  for  drayage  and  storage.     The  net,  or  prime 

cost  of  the  good  is  $ ;  the  total  charges  are  $ .     The  principal  must 

pay  the  agent  in  full  settlement,  $ . 

3.  If  buyer  and  seller  had  dealt  direct,  in  example  2,  who  would  have 
saved  money?     How  much? 

Illustration.     CD.  Rand  sends  Robert  Gates  a  draft  for  $3000  with 
orders  to  invest  in  Greening  apples,  market  quotation  $2.40.     Com.  5%, 
drayage  20c  bbl.,  storage  10c.     Find  all  values. 
Solution. 

$2.40  net  cost  per  bbl.  , 

(5%  of  $2.40)     .12  com.  The  number  of  bbl.  = gross  cost 

-_  J  gross  cost  per  bbl. 

.20  drayage 

.10  storage  1063  =  no.  bbl. 

$2.82  gross  cost  per  bbl.  2.82)3000 

1063  X  $2.40  =  $2551.20  prime  cost  282 

1063  X      .12  =      127.56  com.  1800 

1063  X      .20  =      212.60  drayage  1692 

1063  X      .10  =      106.30  storage  1080 

$2997.66  gross  cost  846 

$3000  -  $2.34  (surplus)  =  $2997.66,  gross  cost.  2.340 

The  principal  receives  1063  bbl.  and  $2.34  surplus. 
The  agent  receives  $3000,  returns  $2.34,  keeps  $127.56  com.  and  pays 
$2551.20  for  the  apples,  $212.60  for  drayage,  and  $106.30  storage. 

ORAL    EXERCISE. 

(Memoranda  of  Purchases.) 
(a)  8000  bu.  oats  @  40c,  commission  l/8c. 
(5)  200  sh.  stock  @  $80,  brokerage  $1/8  per  sh. 

(c)  2000  lb.  lard  @  10c,  com.  2c  per  100  lb.,  misc.  charges  $5. 

(d)  80  A.  land  @  $50,  com.  4%,  transfer  and  survey  $35. 

1.  Compute  agent's  commission  in  each  case. 

2.  Determine  amounts  paid  and  received  by  each  party. 


AGENCY.  251 

Extract  from  Quotation  List. 

Chicago  markets  yesterday: 

Wheat       Opening.     Highest.    Lowest.  Closing. 

July 89i  .90i  .89  .90^ 

Sept.. 91  .921  .901  .92| 

Dec 94|  .961-1        .94f  .96^ 

May 991  1.01  .99i  1.00^ 

Com 

July 51|  .52^  .511  .52^ 

Sept 51f  .521  .51f  .52f 

Dec 48^  .491  .48^  .49^ 

May 50i  .50|-.51  .50  .50| 

Oats. 

July 42  .43  .41i  .43 

Sept 371  .381  .37f  .38| 

Dec 38^  .39i  .38|  .39i 

May 401  .41  .40  .40|-.41 

Pork.. 
Sept 16.55  16.55         16.52         16.52 

EXERCISE. 

(See  Quotation  list  where  necessary.) 

1.  Invest  $5000  in  July  wheat,  opening  quotation,  brok.  l/8c — no 
fraction  of  100  bu.  to  be  bought.     Number  of  bushels? 

2.  What  is  the  purchasing  power  of  $2000  in  September  com,  at 
highest  quotation,  brok.  l/4c. 

3.  Determine  gross  cost  of  500  bbl.  pork,  lowest  quotation,  com. 
2|c  a  bbl. 

4.  An  agent,  charging  4^%  com.,  draws  on  me  for  what  sum  to 
cover  the  full  cost  of  200  cr.  tomatoes — $2.25  expressage  and  drayage 
40c  each? 

Using  account  sales  form  as  a  model,  prepare  an  accoimt  purchase  for 
this  transaction. 

5.  My  draft  for  $700  will  pay  for  how  many  crates  of  eggs,  30  doz. 
each,  @  18c  a  doz.;  com.  4%;  drayage  25c? 

6.  A  receives  from  his  principal  a  draft  for  $12,000  with  orders  to 
secure  three  mgnths'  options  on  as  many  acres  of  certain  coal  lands  as 
possible.  A  allows  $50  for  travelling  expenses  and  4%  for  commission. 
He  secures  options  on  80  A.  at  15  %  of  the  selling  price,  which  is  $420  per  A. 


252 


BUSINESS   ARITHMETIC. 


The  balance  he  spends  m  the  purchase  of  12%  options  on  land  priced  at 
$210  per  A.     On  how  many  acres  of  this  latter  tract  can  he  secure  options? 

7.  I  bought  through  an  agent  60  cr.  choice  tomatoes  @$2.75;  com.  4%; 
expenses  $20.  I  sold  25  crates  direct  @  $3.50  and  the  balance  through  an 
agent  at  $3.80  on  5%  com.     Find  net  returns. 

8.  In  accordance  with  orders,  an  agent  disposes  of  200  A.  of  a  tract 
of  280  A.  @  $68,  and  the  balance  @  $43,  investing  the  net  proceeds  in 
options  on  timber  lands  @  $10  per  A.  His  commission  being  4%  each  way 
and  his  expenses  $50,  he  can  secure  options  on  how  many  acres? 

9.  An  advertising  soUcitor  agrees  to  secure  advertising  for  a  book- 
let, on  25%  commission.  He  secures  35  full  page  advertisements 
@  $11.50  per  inch.  The  pages  are  two  column  width  and  10  inches  depth 
of  column.     His  total  commission  is  $ . 

10.  Extend  this  statement  of  a  real  estate  agent  to  his  principal,  charging 
5%  on  collections. 

Washington,  D.  C,  Feb.  2,  191-. 
Mr.  James  Parker, 

1382  13th  Street,  N.W. 
Dear  Sir 

Enclosed  find  my  check  per  statement  below. 

Respectfully    Robert  Crane,  Agent. 


Tenant 

Premises 

Explanation 

Receipts 

Charges 

C.  P.  Lewis 

1491  N.St.  N.W. 

Rent  to  Feb.  1, 19— 
Repairs  to  range 

60 

— 

3 

60 

James  Farmer 

1280  L  St. 

Rent  to  Feb.  1, 19— 
Plumbing  repairs 
Commission  5% 
Balance  due 

85 

50 

7 

35 

PROFIT  AND  LOSS. 
EXERCISE. 
(Solve  mentally  when  possible.) 
Determine  the  rate  of  gain  or  loss  on : 

1.  500  lb.  Rio  coffee  bought  @  8c,  on  5%  com.,  and  sold  @  9c  direct. 

2.  Quinces  bought  @  $2  and  sold  @  $3  on  5%  commission. 

3.  Determine  the  profit  or  loss  on  1000  bu.  com  bought  @  70c  and 
sold  @  71^c;  brokerage  l/4c  each  way. 

4.  Spring  bran  bought  @  $19.60  per  ton,  com.  5%  and  drayage  $1.00 
per  ton,  must  be  retailed  at  what  rate  to  yield  15%  profit? 


AGENCY.  253 

5.  What  per  cent,  of  profit  is  obtained  by  selling  @  $3.20  apples 
bought  through  a  commission  merchant  @  $2.50,  on  5%  commission? 
Expenses  of  sale  equal  5%  of  the  selling  price. 

6.  What  profit  is  realized  in  retailing  @  $1.50,  through  an  agent 
charging  20%  commission,  a  subscription  book  that  costs  92  cents  to 
publish? 

7.  An  article,  costing  $1.80  to  manufacture,  is  to  be  sold  at  a  price 
to  yield  the  manufacturer  20%  profit,  after  allowing  selling  agents  25% 
commission.     Find  list  price. 

8.  A  sewing  machine  costs  $11.90  to  manufacture.  Find  marked 
price  to  allow  for  25%  profit  and  20%  commission  to  agents. 

9.  How  much  must  the  sales  of  an  agent  increase  to  make  it  profi- 
table to  lower  his  rate  of  commission  from  5|%  to  4^%? 

10.  A  book  costing  90  cents  to  publish  is  sold  @  $1.50  through  agents 
who  are  paid  30%  commission.  How  is  the  publisher's  rate  of  profit 
affected  by  lowering  the  commission  to  20%? 

REVIEW    DRILL    EXERCISE. 

1.  Buy  40  bbl.  @  $5,  com.  5%. 

Find:  gross  cost;  prime  cost;  commission;  sum  agent  pays  seller;  sum 
agent  charges  principal;  sum  agent  earns;  largest  sum  involved.  Repeat, 
if  commission  rate  is  (1)  8c.  bbl.;  (2)  1/4%. 

2.  Sell  2000  bu.  @  60c,  com.  l/4c,  storage  l/2c. 

Find:  sum  buyer  pays;  sum  agent  earns;  sum  handled  by  agent;  net 
proceeds;  the  cost  of  the  sale  to  the  principal. 

3.  State  the  values  that  may  be  computed  from  these  facts.  For- 
mulate and  solve  simple  problems  involved  in  them.  State  whether 
purchase  or  sale. 

(a)  Commission,  $4;  gross  proceeds,  $40. 
(6)  Commission,  $12;  net  proceeds,  $188. 
(c)  Commission,  $60;  prime  cost,  $2000. 
{d)  Commission,  $90;  quantity  400  bbl. 
(e)  5%  commission  on  a  sale  is  $22. 

(/)  An  agent  sends  his  principal  $388  after  reserving  $12  for 
himseK. 

4.  Give  illustrations  showing  what  values  can  be  determined  from: 

(a)  The  commission  and  a  "cent"  rate. 
(6)  The  commission  and  a  rate  per  cent. 
(c)  Number  of  articles  and  price. 
(d)  The  net  and  gross  proceeds. 


254  BUSINESS  ARITHMETIC. 

(e)  The  sum  for  which  an  agent  sells  gdods  and  the  sum  he  remits 
his  principal. 

5.    Give  illustrations  showing  from  what  known  values  the  following 
may  be  computed: 

(a)  The  commission  on  a  sale;  on  a  purchase. 

(6)  The  sum  an  agent  receives  in  making  a  purchase. 

(c)  The  sum  the  principal  receives  from  a  sale. 

(d)  The  gross  cost. 

(e)  The  sum  paid  by  an  agent  in  making  a  purchase;  a  sale. 


CHAPTER  XXXIII. 

INSURANCE. 

To  Pupils.  Collect  and  bring  to  class,  for  reference,  old  fire,  health, 
accident  and  life  policies,  insurance  folders,  and  magazine  advertisements 
that  give  insurance  information. 

INTRODUCTORY    EXERCISE. 

1.  What  is  your  own  definition  of  "insurance"? 

2.  Into  what  two  classes  can  you  divide  all  the  classes  of  insurance? 

3.  Name  several  kinds  of  personal  and  of  property  insurance. 

4.  Give  five  reasons  for  taking  out  life  insurance. 

5.  Give  other  uses  for  life  insurance  than  simply  to  produce  a  fixed 
sum  at  death. 

6.  Read  over  a  life,  accident  or  health  policy.  Are  any  conditions 
imposed  on  the  person  insured,  as  to  occupation,  place  of  residence,  etc.? 
Why  should  such  conditions  be  made? 

7.  Give  several  reasons  for  insuring  one's  property. 

8.  Distinguish  between  general  and  special  property  insurance. 

9.  Study  a  fire  insurance  poHcy  and  name  at  least  eight  conditions 
imposed  on  the  insured.     State  good  reasons  for  these  conditions. 

10.  State  several  ways  in  which  one  might  "forfeit"  his  poHcy. 

11.  Does  property  insurance  protect  the  insured  property  against 
damage?  What  does  it  do?  Has  an  insurance  comp  ny  a  right  to  make 
repairs  rather  than  pay  damages? 

12.  How  do  strict  policy  conditions  tend  to  prevent  losses? 

13.  Why  do  insurance  rates  vary  on  different  kinds  of  property? 

14.  Why  should  a  person  always  read  a  policy  carefully  before  taking 
out  insurance  under  it? 

A  contract  is  an  agreement  between  two  or  more  parties  to 
do  or  not  to  do  a  particular  thing.  It  is  evident  that  the 
parties  must  agree  as  to  the  thing  desired.  It  is  generally 
considered,  also,  that  there  must  be  proper  consideration — that 

255 


256  BUSINESS   ARITHMETIC. 

is,  a  fair  return  in  service  or  value,  for  what  is  received. 
Naturally,  contracts  may  be  of  most  varied  kinds.  Among 
the  commonest  are  those  for  the  sale  or  rental  of  property, 
for  the  payment  of  money,  for  employment  and  for  insurance. 
Contracts  in  the  form  of  notes,  bonds,  etc.,  have  been  touched 
on  elsewhere. 

Contracts  for  insurance  are  agreements  whereby  one  party 
agrees  to  pay  another  party  a  specified  sum  of  money  in  the 
event  of  certain  happenings,  such  as  the  death  or  injury  of 
specified  persons,  or  the  destruction  or  loss  of  property  through 
fire,  water,  theft  or  accident. 

The  written  or  printed  document  containing  the  terms  is  the 
insurance  policy.  The  face  of  the  policy  is  the  sum,  or  limiting 
sum,  which  the  insurer  agrees  to  pay.  The  premium  is  the 
payment  made  by  the  insured  to  the  insurer,  for  protection. 
Payments  of  premiums  after  the  first  period  are  termed 
renewals. 

Life  insurance  policies  run  for  a  term  of  years,  or  for  the 
life  of  the  insured;  health  and  accident  insurance  poUcies 
generally  for  one  year;  transit  insurance  (on  goods  being  trans- 
ported) generally  for  the  period  of  .travel;  fire  insurance  for 
terms  of  one,  two,  three  or  five  years. 

The  rates  of  premium  on  personal  insurance  are  usually 
expressed  in  dollars  per  thousand;  on  property  insurance  in 
dollars  per  $100  or  $1000,  or  as  a  per  cent,  of  the  face  of  the 
policy. 

The  computations  involved  in  the  general  offices  of  insurance 
companies  are  exceedingly  involved  and  require  a  high  degree 
of  mathematical  knowledge.  From  the  standpoint  of  the 
insurance  clerk  and  the  insured,  the  computations  are  very 
simple  and  require  no  new  principles. 

It  is  evident  that  insurance  is  naturally  divided  into  two 
great  groups — personal  insurance  and  property  insurance. 
These  will  be  noticed  briefly. 


INSURANCE.  257 

PERSONAL  INSURANCE. 

Besides  general  life  insurance,  there  are  many  classes  of 
personal  insurance,  of  which  the  following  are  well  known: 

Accident  Insurance  policies  provide  for  payment  to  the 
insured,  or  to  specified  beneficiaries,  of  sums  varying  with 
the  character  and  severity  of  the  injury  to  the  insured.  There 
are  provisions  for  payments  of  surgeons'  fees.  Payments 
may  be  made  in  a  lump  sum,  or  weekly  for  a  specified  number 
of  weeks. 

Health  Insurance  may  be  written  in  connection  with  acci- 
dent insurance,  or  as  a  separate  contract.  One  form  pro- 
vides for  weekly  or  monthly  indemnity  for  illness  arising  from 
a  limited  number  of  diseases;  others  are  much  more  strict 
in  limitations.  The  contract  usually  provides  for  payments  for 
a  period  not  greater  that  twenty-six  weeks  and  for  a  specified 
general  payment  in  case  of  permanent  disability. 

Liability  Insurance  is  written  for  employers  to  cover  settle- 
ments with  employees  for  injuries  for  which  the  employer  is 
responsible.  A  policy  covering  all  or  certain  classes  of  em- 
ployees in  a  manufacturing  plant,  for  example,  is  paid  for 
by  a  premium  based  on  the  total  pay  roll.  The  insurance 
company  defends  for  the  insured  any  damage  suits  brought 
by  employees  on  account  of  injuries,  and  settles  all  legal  or 
just  claims. 

Automobile  liability  insurance  is  also  written  for  any  owner 
of  an  automobile  to  indemnify  him  for  liability  on  account  of 
injuries  he  inflicts  on  others  while  running  his  machine. 

Fidelity  and  Surety  Insurance.  Fidelity  insurance  is  written 
to  protect  the  insurer  against  loss  through  employees  who  do 
not  faithfully  perform  their  duties.  It  covers  the  bonding  of 
employees  who  handle  money,  and  is  often  paid  for  and  taken 
out  by  the  ones  bonded.  Surety  insurance  involves  also  the 
issuing  of  general  court  bonds,  and  bonds  for  the  faithful 
performance  of  any  service  or  contract.  Here  the  insurance 
18 


258  BUSINESS  ARITHMETIC. 

company  may  demand  from  the  person  it  bonds  not  only  a 
premium  but  also  certain  approved  securities  on  which  it  can 
realize  funds  with  which  to  complete  the  contract  in  case  of  the 
failure  of  the  bonded  person  or  company. 

Credit  Insurance  is  in  a  sense,  also,  a  class  of  property 
insurance.  This  is  issued  to  merchants  to  protect  them  from 
excessive  loss  from  bad  debts.  Merchants  who  sell  on  credit 
and  desire  such  insurance  submit  their  books  to  the  credit 
insurance  company  which  averages  the  losses  from  bad  debts 
for  a  series  of  preceding  years.  A  premium  rate  is  then  fixed, 
in  return  for  which  the  insuring  company  agrees  to  make  good 
any  loss  from  bad  debts  during  the  year,  in  excess  of  the 
average  loss.  The  credit  insurance  company  places  restric- 
tions on  the  merchant,  however,  to  prevent  the  careless  grant- 
ing of  credit  to  unreliable  customers. 

EXERCISE. 

1.  Compute  the  cost  of  a  combined  accident  and  health  policy  for 
$7500,  if  the  premium  is  $5.60  per  unit  of  $500. 

2.  What  should  the  insurance  company  pay  the  holder  of  the  above 
policy  for  a  serious  illness  of  9  weeks  3  days,  at  the  rate  of  $37.50  per  week? 

3.  Compute  the  cost  of  a  $75,000  bond  for  a  cashier,  secured  from  a 
bonding  company  at  a  rate  of  $2.45  per  $1000? 

4.  The  pay  roll  of  a  manufacturing  company  amounts  to  $175,000 
per  annum.  A  liability  insurance  company  agrees  to  assume  the  em- 
ployer's liability  for  this  force  at  an  annual  premium  of  3i%.  What  is 
the  premium? 

5.  What  does  the  insurance  company  gain  in  the  above  case,  if  it  pays 
damages  of  $4580  and  incurs  legal  expense  in  defense  and  examinations 
amounting  to  $547.50? 

6.  A  credit  insurance  company  examines  the  books  of  a  mercantile 
house  and  finds  the  normal  loss  from  bad  debts  to  be  $4500.  It  issues 
a  policy  for  an  excess  loss  of  $8000,  at  a  premium  of  4%.  What  is  the 
premium?  For  the  year  covered  by  the  policy  the  loss  is  $7896.  What 
sum  does  the  insurance  company  pay? 

7.  Compute  the  cost  of  a  $7500  excess  policy  at  3J%.  Compute 
the  normal  loss  in  this  case  if  the  respective  losses  for  the  six  preceding 
years  was  $2512,  $2681,  $2512,  $2800,  $3126,  and  $2107. 


INSURANCE.  259 

Life  and  Endowment  Insurance.  Insurance  on  the  life  of  a 
person  is  issued  under  policies  of  most  varied  form.  The  most 
common  classes  might  be  termed  straight  life,  limited  payment 
life,  endowment  and  term.  Life  policies  guarantee  the  pay- 
ment of  a  specified  sum  at  death  of  the  insured,  to  the  bene- 
ficiary named  in  the  policy,  in  consideration  of  premiums  paid 
during  the  life  of  the  insured,  or  during  a  given  period.  Ten- 
payment  and  twenty-payment  life  policies  are  common.  Term 
policies  limit  the  guarantee  payment  in  case  of  death  within  a 
fixed  number  of  years.  Endowment  policies,  in  consideration 
of  annual  premiums  paid  for  a  fixed  number  of  years,  guar- 
antee a  fixed  payment  to  the  beneficiary  at  death  of  the  in- 
sured, or  to  the  insured  at  a  certain  age. 

Premiums  are  usually  expressed  on  a  basis  of  $1000  of 
insurance,  and  vary  with  the  age  of  the  insured  and  the 
character  of  the  insurance  agreement.  They  are  most 
carefully  compiled  and  are  based  on  the  chances  of  life  and 
the  earning  power  of  the  premiums  or  investments  of  the 
company.  The  processes  of  computation  are  exceedingly 
complicated.  ^ 

The  reserve  of  an  insurance  company  is  the  portion  of  income 
from  premiums  that  it  is  required^  by  law  to  set  aside  as  a 
fund  for  the  payment  of  policies  when  they  fall  due.  The 
surplus  of  a  company  is  the  excess  of  assets  over  liabilities. 
In  the  case  of  mutual  companies,  or  those  issuing  participating. 
policies,  a  certain  portion  of  the  surplus  is  returned  to  policy- 
holders whose  contracts  entitle  them  to  share  in  it. 

Many  causes  may  lead  or  force  a  policyholder  to  give  up  his 
policy,  or  to  cease  paying  premiums  on  it.  (Name  some.) 
However,  as  a  rule,  if  he  has  paid  two  or  more  annual  pre- 
miums, he  receives  some  return  for  premiums  already  paid. 
Thus,  he  may  get  cash,  on  surrendering  his  policy,  equal  to 
practically  its  reserve  value.  He  may  secure  a  smaller  policy 
in  exchange,  all  "paid  up,''  for  .whatever  amount  the  reserve 


260  BUSINESS  ARITHMETIC. 

will  purchase;  or  he  may  secure  extended  insurance  for  a 
certain  number  of  years.  Many  companies  will  also  lend 
money  to  the  insured,  on  his  policy,  up  to  its  cash  value. 
Some  companies  allow  a  fixed  number  of  days  of  grace,  say 
thirty,  for  the  payment  of  premium  after  due.  Some  rein- 
state, after  lapse  of  a  payment,  on  proof  of  insurability  and 
payment  of  back  premiums  with  interest. 

A  good,  modern  twenty-year  endowment  policy,  for  example, 
at  age  35  years,  for  $2000,  in  a  certain  company,  if  the 
insured  live,  will  cost  $100.72  per  year  for  20  years,  at  the 
end  of  which  time  it  will  produce  to  the  insured  $2000  and 
its  share  of  surplus  or  accumulation. 

Statement  of  Guaranteed  Values. 


No.  of 

Paid  up 

Upon  Surrender. 

lual  Prem. 

Cash  or  Loan. 

Insurance. 

Participating  Term. 

Paid. 

Value. 

(Participating.) 

Insurance  for 

2 

$112 

$180 

6yr. 

21  days. 

3 

186 

292 

10 

259 

4 

262 

400 

14 

258 

5 

342 

■    512 

15 

0 

6 

424 

•  623 

14 

0 

7 

508 

724 

13 

0 

8 

596 

828 

12 

0 

9 

688 

932 

11 

0 

10 

784 

1032 

10 

0 

11 

884 

1136 

9 

0 

12 

988 

1236 

8 

0 

13 

1096 

1336 

7 

0 

14 

1210 

1432 

6 

0 

15 

1328 

1528 

5 

0 

16 

1448 

1620 

4 

0 

17 

1572 

1722 

3 

0 

18 

1762 

1804 

2 

0 

19 

1838 

1892 

1 

0 

20 

2000  and 

accumulations, 

Bhaxes   in 

dividends,    ii 

addition. 


INSURANCE.  261 

EXERCISE. 

Assume  that  the  above  policy  was  taken  out  on  July  17,  1910. 

1.  What  sum  could  be  borrowed  on  the  policy  at  the  end  of  7  years? 
On  August  18,  1917? 

2.  If  the  insured  surrenders  the  policy  on  May  14,  1916,  what  amount 
of  paid  up  insurance  should  he  receive? 

3.  What  sum  can  he  borrow  on  the  policy  after  12  premiums  have  been 
paid? 

4.  If  the  insured  surrenders  the  policy  on  May  19,  1918,  he  can  secure 
for  it  insurance  for  what  term?  How  much  is  paid  then  in  case  of  death 
6  months  later,  no  allowance  being  made  for  accumulations? 

5.  In  1912  his  dividend  was  $19.66.  He  allowed  it  to  go  to  reduce 
his  premium.     What  was  his  net  premium  in  this  year? 

6.  What  is  the  corresponding  premium  per  annum  for  a  policy  of 
$17,000? 

7.  What  cash  surrender  value  has  this  new  policy  after  9  annual 
premiums  have  been  paid? 

8.  What  is  the  premium  on  a  $12,000  poUcy  at  same  rate?  If  the 
insured  dies  after  the  payment  of  five  annual  premiums,  what  sum  do  the 
heirs  receive?  What  sum  in  excess  of  premiums  paid  in?  What  sum  in 
case  accumulations  during  the  period  of  insurance  have  amounted  to 
$1,345.61? 

Annual  Premium  Rates  for  Different  Types  of  Policies. 
Per  $1000  of  Insurance.  , 

Ordinary.      10  Pay.       20  Pay.  10  Year  20  Year 

Age.  Life.  Life.  Life.  Endowment.        Endowment. 

21  $18.50  $46.40        $28.30  $101.68  $47.70 


25 

20.26 

49.34 

30.17 

102.00 

48.10 

28 

21.73 

51.82 

31.76 

102.29 

48.47 

30 

22.90 

53.62 

32.92 

102.52 

48.78 

35 

26.48 

58.68 

36.27 

103.25 

49.80 

40 

31.12 

64.69 

40.43 

104.33 

51.43 

EXERCISE. 

1.  Find  the  cost  of  a  poHcy  of  each  kind  for  $4500  issued  at  age  28. 

2.  A  man  at  age  30  takes  out  a  20-payment  life  policy.  When  he  has 
completed  payments  on  it,  how  much  less  than  the  face  of  the  policy  has 
he  paid  in? 

3*    WTiy  does  the  age  of  the  insured  affect  the  amount  of  the  premium? 


262  BUSINESS   ARITHMETIC. 

4.  Give  general  reasons  for  the  difference  in  amount  of  premium  for 
policies  of  different  kinds,  at  any  fixed  age. 

5.  Suppose  a  man,  at  age  21  years,  had  taken  out  a  $5000  policy  of 
each  of  the  above  kinds.  Suppose  that  he  died  at  the  age  of  50  yeajB. 
What  benefit  would  he  have  had  from  each  policy  during  his  life?  What 
benefit  would  his  heirs  receive  in  each  case? 

6.  Why  is  an  insurance  company  able  to  pay  large  sums  at  death 
of  the  insured  when  it  receives  only  small  premiums? 

7.  Name  several  advantages  of  insuring  at  as  early  an  age  as  possible. 

8.  Why  are  intending  poUcyholders  required  to  undergo  a  strict  medical 
examination? 

Income  Insurance.  In  return  for  the  payment  of  annual 
premiums  varying  as  usual  with  the  age  of  the  insured,  the 
insurance  company  agrees  to  pay  the  beneficiary,  on  the 
death  of  the  insured,  a  fixed  income  per  month,  for  a  fixed 
number  of  years,  or  for  life,  in  place  of  paying  a  lump  sum. 
Under  some  policies  both  a  lump  sum  and  an  income  is  pro- 
vided. If  the  income  is  payable  during  the  lifetime  of  the 
beneficiary,  then  the  premium  rate  depends  in  part  on  the  age 
of  this  person.  Income  insurance  can  be  arranged  in  income 
multiples  of  $10. 

Old  Age  and  Savings  Insurance.  This  insurance  is  increasing 
in  this  country  and  is  common  abroad.  For  the  payment  of 
small  monthly  premiums,  ordinary  straight  life  and  endow- 
ment policies  are  issued.  There  are  also  combined  insurance 
and  annuity  policies,  involving  payment  in  full  at  death  before 
65,  years  of  age,  and  in  full,  less  annuities  paid,  for  death 
between,  say,  65  and  69.  Annuities  may  be  paid  after  65 
years,  the  annual  payment  being  a  certain  fraction  of  the  face 
of  the  policy.  Straight  old  age  pension  policies  are  also  issued, 
wherein  a  certain  annuity  is  given  after  the  age  of  60  or  65,  in 
return  for  annual  premiums  paid  up  to  that  age.  In  case  of 
death  before  the  annuity  age  begins,  the  amount  of  the 
premiums  paid  in  is  turned  over  to  the  heirs.  The  maximum 
annuity  is  often  a  small  sum,  say  $200,  and  the  maximum 


INSURANCE.  263 

insurance  limit  is  very  small.     No  medical  examination  is 
required  for  old  age  pensions. 

EXERCISE. 

1.  Compute  the  annual  cost  of  an  old  age  pension  policy  ensuring  an 
annuity  of  $200,  purchased  at  an  age  of  36  yr.,  at  the  rate  of  $3.14  per 
month.  How  much  will  have  been  paid  in  in  premiums  by  the  time  the 
insured  reaches  the  age  of  60? 

2.  Anderson  takes  out  an  old  age  policy  guaranteeing  $400  annuity 
after. the  age  of  60  years,  and  the  face  of  the  premiums  in  case  of  death 
at  an  earher  age.  He  is  34  years  old  and  pays  a  monthly  premium  of 
$1.12  per  $100  annuity.  Compute  his  monthly  and  yearly  premiums. 
In  case  of  the  death  of  the  insured  at  the  age  of  57  years,  what  sum  should 
be  paid  his  beneficiaries? 

3.  Roberts,  aged  26,  takes  out  a  $400  savings  endowment  policy  payable 
in  twenty  years  or  at  death.  What  is  his  monthly  premium,  if  the  rate 
is  $2.19  per  $500  insurance?     What  is  his  annual  premium? 

4.  Newton,  aged  40  years,  takes  out  an  income  policy,  guaranteeing 
a  monthly  income  of  $60  at  his  death,  to  his  beneficiary  for  20  years,  at  a 
premium  rate  of  $5.76  per  $10  income  unit.     What  is  his  annual  premiimi? 

5.  Osgood,  aged  30  years,  takes  out  an  income  policy,  calling  for  the 
payment  to  his  beneficiary  after  his  death,  of  a  monthly  income  of  $75 
for  20  years,  at  a  premium  rate  of  $33.47  per  $10  income  imit.  What  is 
his  annual  and  his  quarterly  premium?  In  case  of  death,  at  age  47,  what 
will  he  have  paid  in  premiums?  What  siun  will  his  beneficiaries  receive 
during  the  following  20  years?  Why  is  the  insurance  company's  loss 
less  than  the  difference  between  these  two  amounts? 

PROPERTY  INSURANCE. 
EXERCISE. 

1.  Is  death  sure,  sooner  or  later,  to  every  insured  person? 

2.  Is  loss  by  fire  certain  to  occur,  sooner  or  later,  to  every  building 
insured  against  it?     Is  lightning  sure  to  strike  every  building? 

3.  Name  differences  between  life  and  property  insiu-ance,  as  regards 
probable  losses  payable  by  insurance  companies? 

4.  Why  are  property  insurance  rates  lower  than  personal  insurance 
rates? 

5.  Why  should  fire  insurance  rates  vary  with  the  (a)  use  of  a  building; 
(&)  with  its  surroundings;  (c)  with  the  character  of  material  and  construction 
of  it? 


264  BUSINESS   ARITHMETIC. 

6.  Why  is  a  brick  paper  mill  a  greater  "risk"  than  a  brick  residence^ 

7.  Why  do  property  insurance  companies  make  restrictions  as  to  how 
an  insured  building  shall  be  built  or  used? 

8.  Why  do  fire  insurance  companies  favor  and  work  for  strong  fire 
departments? 

9.  Why  are  country  dwellings  insured  at  a  higher  rate  than  city 
dwelhngs  of  the  same  class? 

10.  Read  the  conditions  of  a  policy  and  determine  some  of  the  duties 
of  the  person  whose  property  is  insured,  in  case  of  damage  to  it  which  is 
covered  by  insurance. 

As  in  the  case  of  life  insurance  companies,  property  insur- 
ance companies  may  be  either  stock  companies,  owned  and 
managed  for  the  stockholders'  profit,  under  certain  legal 
restrictions;  or  mutual  companies,  in  which  the  policyholders 
share  in  gains  and  losses.  Some  companies  are  partially 
mutual. 

Policies  vary  greatly  in  form  and  conditions.  They  may  be 
valued  policies,  stating  a  fixed  amount  to  be  paid  in  case  of  loss; 
or  open  policies,  in  which  a  maximum  payment  is  named  but 
the  actual  payment  is  based  on  evidences  of  loss. 

Illustrations.  Insurance  on  ships  is  frequently  covered  under  valued 
policies  Insurance  on  cargoes,  and  ordinary  fire  insurance  is  generally 
covered  by  open  policies.  In  case  of  total  or  partial  loss  under  open  policies, 
the  insurance  company  pays  that  part  of  the  maximum  stated  in  the  policy 
that  will  indemnify  the  insured.  Thus,  if  $2500  damage  is  caused  by  fire 
to  a  building  insured  for  $6000,  the  entire  damage  is  paid;  but  if  the  damage 
is  $6800,  only  the  maximum  of  the  policy,  $6000,  is  paid. 

Some  policies  contain  a  co-insurance  clause  requiring  that 
the  property  shall  be  kept  insured  up  to  a  certain  per  cent,  of 
its  value — otherwise  that  per  cent,  only  of  the  loss  is  paid  that 
the  insurance  named  is  of  the  per  cent,  required. 

Illustration.  If  80%  is  the  per  cent,  required  to  be  insured  then,  on 
a  $15,000  building,  for  example,  $12,000  insurance  must  be  carried  to 
receive  full  benefit.  In  case  of  $10,000  insurance,  five-sixths  of  $12,000, 
only  five-sixths  of  any  loss,  up  to  the  face  of  the  policy  will  be  paid. 

Some   policies   contain   an   average   clause,   specifying   the 


INSURANCE.  265 

payment  of  that  part  of  the  loss  that  the  insurance  is  of  the 

value  of  the  property. 

Illustration.  If  the  insurance  on  a  piece  of  property  is  four-ninths 
of  its  value,  four-ninths  of  any  loss  will  be  paid. 

Insurance  companies  reserve  the  right  of  replacing  or  re- 
pairing damaged  property  covered  by  their  policies,  in  place 
of  paying  cash  for  losses.  In  case  the  property  is  insured  in 
several  companies,  the  loss  is  usually  apportioned  among  them 
according  to  the  faces  of  the  policies  they  issued  on  it.  Some- 
times a  single  company  issues  a  policy  on  a  valuable  property, 
for  a  heavy  amount,  and  re-insures  a  part  of  its  "risk"  in 
another  company.  Losses  are  appraised  by  special  inspectors 
and  often  agreements  are  reached  with  owners,  as  to  the  loss 
payable. 

The  rate  of  premium  varies  with  the  character  of  the  risk 
and  the  period  of  the  policy.  It  may  be  for  a  specified 
number  of  days,  for  a  year  or  for  a  period  of  years,  or  in 
the  case  of  merchandise  in  transit,  for  the  length  of  a  certain 
voyage.  "Short  rates"  are  often  charged  for  terms  under  one 
year,  and  are  relatively  higher  than  the  yearly  rates.  Rates 
are  expressed  in  cents  per  $100,  or  as  a  rate  per  cent. 

Many  of  the  computations  of  premiums  and  settlements  of 
losses  are  based  on  the  simple  principles  of  ratio  and  of  per- 
centage. No  new  process  is  involved  in  any  problems  in  this 
book. 

SOME  CLASSES  OF  PROPERTY  INSURANCE. 

Fire  Insurance  is  insurance  of  property  against  loss  by  fire. 
The  liability  covers  damage  by  water  in  fighting  the  fire,  or 
destruction  of  property  in  the  same  cause.  Sometimes,  it 
covers  damage  caused  by  Hghtning,  or  by  wind  storms. 

Marine  Insurance  is  insurance  of  vessels  and  cargo  against 
the  perils  of  navigation.  This  includes  damage  by  fire,  or 
storm,  shipwreck,  etc.  Marine  policies  frequently  contain 
the  average  clause. 


266  BUSINESS  ARITHMETIC. 

Transportation  Insurance  is  insurance  of  goods  in  transit  by 
land,  or  by  land  and  water,  against  loss  resulting  from  the 
ordinary  dangers  of  such  traffic. 

Live  Stock  Insurance  is  issued  to  cover  loss  by  casualties  to 
horses,  cattle,  etc. 

Miscellaneous  Special  Types  of  Insurance.  It  is  said  that  it 
is  now  possible  to  insure  any  property  against  any  damage. 
The  following  titles  illustrate  common  types,  and  are  practi- 
cally self-explanatory:  Automobile  and  equipment  insurance, 
tourists'  (personal  effects)  insurance,  motor  boat  and  equip- 
ment insurance,  steam  boiler  insurance,  explosion  insurance, 
theft  insurance,  plate  glass  insurance,  derailment  insurance, 
fly  wheel  insurance,  etc. 


ORAL 

EXERCISE. 

Find  the  cost  of  insurance: 

Face  of  Policy.        Premium.            Face  of  Policy. 

Premium. 

1.           $  2,000                    U% 

5.      $  4,000 

$1.25  per  $100 

2.          $  3,600                      f  % 

6.      $  3,000 

.80  per  $100 

3.          $50,000                    2i% 

7.       $12,000 

1.75  per  $100 

4.          $  8,000                    11% 

8.       $  7,500 

2.50  per  $1000 

Find  the  rate  of  premium  and  express  in  two  ways: 

Face.       Premium. 

Face. 

Premium. 

9.         $4,000       $120 

11.       $12,000 

$150 

10.        $6,000            45 

12.       $  5,000 

75 

13.     $400  will  purchase  $  ?  insurance  at  2%. 

, 

14.     $600  will  purchase  $  ?  insurance  at  1\%. 

15.    $90  is  the  premium  on  a  poHcy  of  $  ?  at  $1.50  per  $1000. 

EXERCISE. 

1.  Johnson  takes  out  $2500  insurance  on  his  touring  automobile  at 
2%  at  a  cost  of  $ . 

2.  What  will  it  cost  Robinson  to  place  $1500  insurance  on  his  stable 
for  one  year  at  25c  per  $100,  and  $800  on  his  horses  and  carriages  at  40c 
per  $100? 

3.  Newton  takes  out  an  80%  policy  on  his  brick  house,  valued  at  $7500, 
at  a  rate  of  $2.75  per  $1000.     It  costs  him  $ . 


INSURANCE.  267 

4.  A  factory  owner  takes  out  $40,000  insurance  on  his  buildings  at 
3^%;  and  $3000  special  insurance  on  his  fly-wheel,  at  $12.50  per  $1000. 
His  total  payment  is  $ . 

5.  James  Sanborn  insures  his  property  through  a  broker,  as  follows: 
Brick  and  stone  house,  valued  at  $24,000,  at  75%  of  its  value  at  $3.75  per 
$1000;  a  broad  plate  glass  window,  for  $400  at  lf%;  furniture,  valued 
at  $12,500,  at  $5.75  per  $1000  for  five  years;  a  brick  stable  and  garage,  for 
$2000,  at  $2.50  per  $1000  one  year  policy;  horses  and  vehicles,  $2000  at 
$4.25  per  $1000;  automobile  and  equipment,  $2000  at  3^%;  frame  boat 
house  $600,  at  60c  per  $100;  motor  boat  and  equipment,  $1000  at  2f  %. 
Compute  his  total  annual  cost  of  insm'ance. 

6.  A  shipment  of  goods  to  Manila,  valued  at  $85,000,  is  insured  for 
90%  value  at  11%.     What  does  the  insurance  cost? 

EXERCISE. 

1.  On  a  house  insured  for  $7500,  damage  is  caused  to  the  extent  of 
$3748.     What  should  the  insurance  company  pay? 

2.  Randall  insures  his  brick  house,  valued  at  $9000,  at  4/5  value  at  2%. 
The  property  is  damaged  to  the  extent  of  $8500.  What  should  the  insurance 
company  pay?  What  is  his  net  loss  and  that  of  the  company,  allowing  for 
premiums? 

3.  Property  worth  $12,000  is  insured  for  $7500,  under  a  policy  con- 
taining an  80%  clause.  In  case  of  proved  damage  to  the  extent  of  $2750, 
what  should  the  insurance  company  pay? 

4.  On  a  stock  of  goods  amounting  to  $30,000,  insurance  is  carried  as 
follows:  Company  A,  $4000;  Company  B,  $6000;  Company  C,  $15,000. 
The  property  is  damaged  by  fire  to  the  extent  of  $16,500.  What  should 
each  company  pay? 

5.  A  stock  bam,  insured  for  $2500  against  damage  by  tornados  at 
80c  per  $100,  is  damaged  in  such  a  storm  to  the  extent  of  $3750.  What  is 
the  real  loss  of  the  owner? 

6.  Prepare  an  illustrative  example  showing  how  the  losses  paid  to  a 
poUcyholder  are  met  by  other  poUcy holders  in  the  same  company. 

7.  Robinson  bought  merchandise  abroad  to  the  value  of  £3000. 
(£1  =  $4.86).     He  paid  a  commission  of  2%,  and  insurance  of  3/4  value, 

at   11%  (average  clause).     The  entire  shipment  cost  him  $ .     Fire 

broke  out  in  transit  and  the  goods  were  damaged  to  the  extent  of  $5860. 
What  should  the  insurance  company  pay?     What  was  Robinson's  net  loss? 

8.  Fire  broke  out  on  the  estate  of  James  Sanborn  (ex.  5  of  previous 
exercise),  causing  loss  of  $19,200  on  the  house; -destroying  all  but  $800 


268  BUSINESS  ARITHMETIC. 

worth  of  furniture;  destroying  the  plate  glass  window;  damaging  the  stable 
to  the  extent  of  $1500;  destroying  the  boat  house,  and  damaging  the  motor 
boat  to  the  extent  of  $350.  Xhe  insurance  company  repaired  the  motor 
boat,  and  sent  a  check  for  the  other  losses  for  which  it  was  responsible  to 
Mr.  Sanborn.     The  face  of  check  was  $ . 

9.  The  "Warfield,"  freighter,  valued  at  $180,000,  is  insured  under 
average  clause  poUcies,  for  $40,000  in  Company  A  and  for  $65,000  in 
Company  B —  the  former  at  21%  and  the  latter  at  2^%.  (Pre- 
miums?) On  one  trip,  the  vessel  carried  a  shipment  of  28,000  bushels  of 
corn,  valued  at  61c  per  bu.  and  insured  at  1^%  to  cover  goods  and 
charges.  (Face  of  policy  and  premium?)  The  vessel  also  had  aboard 
some  general  merchandise  valued  at  $85,000  and  insured  for  3/4  value  at 
lf%  —  average  clause.  (Premium?)  Fire  broke  out  on  this  trip. 
The  general  merchandise  was  destroyed  or  thrown  overboard.  The 
insurance  company  sold  the  grain  that  was  not  destroyed,  amounting  to 
7520  bu.,  at  31c  per  bu.  The  damage  to  the  vessel  was  appraised  at 
$57,800.  Compute  the  payments  by  each  company,  assuming  that  Company 
B  carried  the  insurance  on  both  grain  and  merchandise.  Compute  the 
net  losses  of  the  companies  and  of  the  owners,  taking  premiums  into 
account. 


CHAPTER   XXXIV. 
TAXATION  AND  PUBLIC  REVENUE. 

GENERAL    INTRODUCTORY    EXERCISE. 

1.  What  protection  does  a  city  government  give  its  inhabitants? 

2.  What  besides  protection  does  it  supply? 

3.  What  does  it  do  for  the  young  people? 

4.  How  does  the  local  government  aid  travel  and  intercourse? 

5.  How  are  your  public  officers  chosen? 

6.  Who  can  vote?     Does  it  cost  anything  to  vote? 

7.  How  IS  the  money  for  local  government  expenses  secured? 

8.  What  kind  of  property  is  taxed? 

9.  Why  must  many  business  men  pay  for  a  license  to  do  business? 

10.  What  does  your  state  government  do  for  its  citizens? 

11.  How  does  it  raise  money  for  these  objects? 

12.  Name  twelve  public  objects  for  which  the  United  States  Govern- 
ment pays  money.     How  are  the  people  benefited  in  each  case? 

13.  How  does  the  National  Government  aid  commerce  and  industry? 

14.  How  does  the  National  Government  raise  its  funds? 

15.  Why  does  the  cost  of  government  increase  steadily? 

16.  In  what  ways  are  governments  doing  more  than  they  used  to  do 
for  their  people? 

1.  LOCAL  AND  STATE  TAXATION. 
The  cost  of  local  and,  to  some  extent,  of  state  governments 
is  met  by  a  tax  or  charge  levied  on  real  estate,  in  proportion  to 
its  value;  and  on  general  personal  or  movable  property — such 
as  furniture,  machinery  not  built  in  as  a  part  of  a  building, 
horses  and  wagons,  and  sometimes,  stocks  and  bonds.  A 
further  income  is  derived  from  fees  for  licenses,  or  permits 
to  conduct   special  businesses;  from  assessments,  or  charges 


270  BUSINESS   ARITHMETIC. 

against  private  real  estate,  for  improvements  in  the  form  of 
sidewalks,  water  mains,  etc.;  and  from  fines  and  'permits. 
Polls  or  capitation  taxes  are  levied  on  all  qualified  voters. 

EXERCISE. 

Secure  a  printed  statement  of  your  own,  or  a  nearby,  city  showing  total 
expenditures  for  a  recent  year  for  governmental  purposes.  Show  what 
are  the  largest  items  of  expense.     Find  the  average  expenditure  per  person. 

The  greatest  source  of  public  income  is  realty.  The  process 
of  taxing  realty  in  one  Eastern  city  is  as  follows:  A  complete 
assessment  (valuation)  of  real  property  "  at  not  less  than  two- 
thirds  of  the  real  value  thereof,"  is  made  every  three  years 
by  tax  assessors.  The  chief  assessor  and  his  assistants  then 
sit  as  a  Court  of  Equalization  and  Review  to  hear  owners  and 
witnesses,  readjusting  any  seemingly  unfair  assessment.  Their 
decisions  are  then  confirmed  by  the  governing  ofiicials. 
Special  assessments  are  made  annually  of  property  that  has 
changed  in  value  $500  or  over,  owing  to  its  improvements,  or 
to  destruction  of  buildings.  The  tax,  1|%  of  assessed  value, 
is  payable  to  the  collector  of  taxes  in  May  for  the  year  ending 
June  30th. 

Tax  rates  on  real  or  personal  property,  expressed  as  mills  • 
or  cents  per  $1.00  or  $100,  or  as  a  rate  per  cent.,  are  based  on 
the  assessed  value.  Thus,  1^  %,  15  mills  (1|  %  of  $1.00) 
or  $1.50  (on  $100)  are  equivalent  rates.  Several  taxes,  such 
as  local,  county  and  state,  may  be  levied  on  the  same  property, 
and  collected  at  one  time. 

ORAL   EXERCISE. 

Express  each  rate  in  two  other  forms: 

1.  1.8%.         3.     li%.  .5.     12  mills  ($1.00).     7.    22^  mills. 

2.  3/4%.         4.     U%.  6.     16  mills.  8.    $1.75  ($100). 
Find  total  rates  equivalent  to  these  series: 

11.  2/5%,  8  mills,  $1.50.        '  13.     U%,  75c,  7J  mUls. 

12.  li%,  3  mills,  $1.25.  14.    75c,  $1.25,  2\%. 


TAXATION   AND   PUBLIC   REVENUE.  271 

The  computation  of  the  tax  involves  simple  percentage  or 
multiplication. 

Illustration.     Compute  state  and  local  tax  at  1|%  and  8  mills,  on 
property  assessed  at  $4200. 

Solution.     State  tax  =  U%  of  $4200  =  $63.00 

Local  tax  =  8  mills  per  $100  =  4200  X  $.008  =  $33.60. 

Note.     The  total  tax  may  be  found  directly.     8  mills  =  4/5%.     1|%  + 
4/5%  =2x\%.     2j%%  of  $4200  =  $96.60. 

ORAL   EXERCISE. 


Find  total  tax. 

Hates. 

Assessed  Value. 

Rates.       Assessed  Va 

1.        2/5% 

$4500 

5. 

8  mi.              $24,500 

2.         U% 

7280 

6. 

1/4%  and           6000 

3.        $1.25 

14,500 

$1.50 

4.            .50c 

8420 

7. 

15m.,  2^m.          8400 

EXERCISE. 

Find  missing  values. 

State  each  example  in  form  of  a  question,  and  show  how  it  differs  from 
preceding  problems  of  a  similar  nature. 

1.  John  Sampson's  property  is  assessed  for  $12,600,  at  11%.     His  tax 

is  $ . 

2.  C.  P.  Crane  has  assessed  at  $1.25  per  sq.  ft.,  a  lot  measuring  240  ft. 
X  80  ft.     His  tax,  at  1^%,  is  $ -. 

3.  Robert  Walter  owns  three  lots,  assessed  at  $8420,  $9200  and  $14,620. 
On  the  latter  is  a  building  assessed  at  $16,500.  He  has  personal  property 
assessed  at  $4500,  but  $1000  is  exempt  by  law.  His  total  tax  at  12|  mills 
is$ . 

4.  A  Permanent  Improvement  Fund  tax  of  2§  mills,  levied  by  a 
western  city  on  property  assessed  at  $26,450,  costs  its  owner  $ . 

5.  A  Current  Expense  Fund  is  created,  in  a  Minnesota  city,  by  a 
tax  of  6/10%.  On  a  realty  assessment  of  $99,547,484  and  a  personal 
property  assessment  of  $29,049,250,  this  would  yield  what  sum  if  every 
person  paid?  In  this  city,  the  delinquent  tax  averages  4|%.  What  net 
tax  should  the  property  yield? 

6.  A  western  city  recently  levied  this  general  tax  per  $1000.  What 
does  it  cost  C.  A.  Mansfield  whose  property  is  assessed  at  $8455? 

State  revenue S1..50 

State  School  and  University  Fund 2.23 


272  BUSINESS  ARITHMETIC. 

County  revenue 2.75 

School  revenue 6.00 

City  funds 14.37 

? 
7.    James  Field,  collector  for  the  town  of  Mayfield,  receives  a  salary 
of  3%  of  his  collections.     If  he  collects  a  $1.25  tax  on  $364,000  of  property, 
he  earns  $ — -. 

Note.  Penalties  are  charged  for  delinquent  (unpaid)  taxes.  In  one 
city,  penalty  interest  of  1%  per  month  is  charged.  In  default  of  payment, 
the  property  is  sold,  but  the  purchaser  gets  simply  a  tax  deed  bearing  1% 
per  mo.  interest,  and  the  owner  may  redeem  his  property  within  two  years. 

8.  The  property  here   shown  is  as- 
120'                            sessed  at  $30  per  front  foot,  and  taxed  8 

mills.  The  tax  due  on  Jan.  1,  was  not 
paid  until  Oct.  1,  incurring  a  penalty,  at 
3/4%  per  month,  of  $ . 

9.  A  tax  certificate  of  $241.60,  bear- 
ing 1%  interest  per  month,  dated  Dec. 
10  last,  is  worth  to-day  $ . 

10.  J.  C.  Parker  leaves  his  wife  by 
will,  $42,000  hi  property;  his  son,  $12,- 
600,  and  a  niece  $5200.     The  state  has 

an  inheritance  tax  of  1?%  on  bequests  to  Imeal  heirs,  and  6%  to  others. 
The  tax  on  these  bequests  yields  the  state  $ . 

11.  The  Randolph  Manufacturing  Co.  is  taxed  1%  of  its  legal  capital, 
on  issuance  of  its  state  charter.  If  the  authorized  capital  is  $240,000, 
the  tax  is  $ . 

12.  An  interurban  trolley  road  is  taxed  90c  per  $100  on  its  equipment 
and  1/2%  on  its  gross  earnings.  Last  year  its  equipment  was  assessed 
at  $8,240,000  and  its  gross  earnings  were  $6,547,325.60.  Its  state  tax 
amounted  to  $ . 

Many  problems  arise  in  tax  offices  when  making  up  advanced 
estimates.     A  few  are  stated  below. 


EXERCISE. 

1.  What  rate  in  mills  and  per  cent,  is  sufficient  to  raise  $12,460  on  an 
assessed  value  of  $800,000? 

2.  In  a  city  having  a  fixed  legal  tax  rate  of  15  mills,  the  total  assessed 
value  must  be  fixed  at  what  sum,  in  order  that  an  income  of  $172,000  may 
be  secured? 


TAXATION   AND   PUBLIC   REVENUE. 


273 


3.  Last  year,  in  a  Nebraska  town,  the  legal  rate  of  3|  mills  yielded 
$142,500.  This  year  the  income  required  being  only  $122,000,  the  assessed 
value  may  be  reduced  to  approximately %. 

4.  A  Pennsylvania  town  must  meet  the  following  appropriations  by 
a  tax  of  ?  mills,  on  an  assessed  valuation  of  $1,246,300:  salaries  $2140; 
schools,  $8720;  roads,  $2465;  general  expenses  $5840.  Assume  5% 
deUnquent  tax. 

5.  520  polls  @  $1.50;  Ucense  fees,  $1260;  and  a  $1.25  tax  on  realty 
and  personalty,  assessed  at  $1,652,000,  will  yield  an  annual  income  of  $ . 

• 
For  convenience  in  calculation,  tax  computation  tables  are 

often  prepared  for  the  total  tax  rate  of  a  town  or  city.     The 

following  table  gives  the  tax,  at  the  given  rate,  on  any  sum 

from  $10  to  $99.     The  tax  on  other  sums  is  found  by  "  pointing 

off  "  and  by  addition.     The  original  table  must  be  computed 

with  great  care,  and  written,  preferably,  in  six  decimal  places. 


7§  mills 


Tax  Table 


Per  $100 


Unit  lines 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1 

.075 

,0825 

.0900 

.0975 

.0150 

.1125 

.1200 

.1275 

.1350 

.1425 

1 

2 

.150 

.1575 

.1650 

.1725 

.1800 

.1875 

.1950 

.2025 

.2100 

.2175 

2 

3 

.225 

.2325 

.2400 

.2475 

.2550 

.2625 

.2700 

.2775 

.2850 

.2925 

3 

4 

.300 

.3075 

.3150 

.3225 

.3300 

.3375 

.3450 

.3525 

.3600 

.3675 

4 

5 

.375 

.3825 

.3900 

.3975 

.4050 

.4125 

.4200 

.4275 . 

4350 

.4425 

5 

6 

.450 

.4575 

.4650 

.4725 

.4800 

.4875 

.4950 

.5025 

.5100 

.5175 

6 

7 

.525 

.5325 

.5400 

.5475 

.5550 

.5625 

.5700 

.5775 

.5850 

.5925 

7 

8 

.600 

.6075 

.6150 

.6225 

.6300 

.6375 

.6450 

.6525 

.6600 

.6675 

8 

9 

.675 

.6825 

.6900 

.6975 

.7050 

7125 

.7200 

.7275 

.7350 

.7425 

9 

0123456789 
Illustrations.     The  tax  on  $82  is  found  on  the  horizontal  line  "8" 

at  its  intersection  with  the  vertical  unit  line  "2."  It  is  $.615  or  62c. 
The  tax  on  $8200  (100X82)  =  $61.50.     The  tax  on  $972  =  tax  on  $900 

+  tax  on  $72,  etc. 


EXERCISE. 

1.     Show  how  the  above  table  may  be  constructed  by  addition, 
how  short  methods  may  be  used  for  construction  or  checking. 

19 


Show 


274  BUSINESS  ARITHMETIC. 

2.  Read  from  the  table,  the  tax  on  assessed  values  of: 

$840  $24,200  $1,220  $    815 

1200  36,450  19,600  2,290 

960  8,000  452  42,600 

3.  Use  the  table  to  read  the  tax  on  these  same  values  at  15  mills. 

4.  Construct  tax  tables  for  these  rates:  (a)  1^%;  (6)  60c;  (c)  $1.45; 
id)  $27.60  (per  $1000). 

Licenses.  Many  municipalities  secure  an  income  from  a 
charge  for  licenses  to  conduct,  certain  businesses.  An  eastern 
city  received  from  this  source,  in  one  year,  $572,473.43. 
These  are  a  few  of  its  rates: 

Busmess.  Rate.                             Remarks. 

Apothecaries $     6.00 per  annum. 

Auctioneers  100.00 per  annum. 

Boarding  houses,  hotels,  public.  .       1.00 per  guest  room. 

Brokers,  real  estate 50 per  annum. 

Carriages,  for  hire 6.00 one  horse. 

Carriages 9.00 more  than  one  horse. 

Circuses 200.00 per  day. 

Confectioners    12.00 per  annum. 

Livery  stables 25.00   per  annum  10  stalls. 

Livery  stables    2.00 each  additional  stall. 

Passenger  transfer  lines 6.00. each  vehicle  seating  ten  or  less. 

Passenger  transfer  lines 12.00. . .  .each  vehicle  seating  over  ten. 

EXERCISE. 

Using  these  license  rates,  tabulate  the  income  derived  by  a  city  from 
61  apothecaries,  18  auctioneers,  529  street  cars,  63  confectioneries,  17 
livery  stables  (under  10  stalls),  8  stables — total  of  286  stalls,  282  carriages 
for  hire;  41  brokers;  21  hotels,  guest  rooms  respectively:  149,  217,  300,  84, 
29,  80,  120,  360,  400,  210,  88,  146,  97,  63,  48,  88,  95,  85,  66,  22,  38. 

Assessments.  In  some  cities  a  portion  of  the  cost  of  public 
improvements  and  of  repairs,  such  as  curbing,  sidewalks,  water 
mains,  sewers,  alleys,  etc.,  is  levied  by  assessment  against 
property  that  is  especially  benefited.  If  an  owner  asks  for 
improvements,  he  must  deposit  his  estimated  share  of  the 
cost  in  advance.  If  he  does  not  ask  for  the  improvements, 
his   payments   may   usually   extend   over   two   years.     The 


TAXATION  AND   PUBLIC   REVENUE.  275 

proportion  paid  b}  private  owner  and  the  methods  of  assess- 
ment vary  greatly  in  different  cities.  Several  different 
methods  are  illustrated  in  the  following  problems. 

EXERCISE. 

1.  John  Evans  is  assessed  $  ?  for  a  water  main,  at  a  rate  of  $1.10  per 
front  foot,  on  two  lots  of  85  ft.  and  40  ft.  frontage  respectively. 

2.  320  ft.  of  curbing  are  laid  at  a  cost  of  90c  per  foot.  The  frontage 
benefited  is  300  ft.,  of  which  Adams  owns  65  ft.  He  is  assessed  one-half 
the  pro  rata  cost,  or  $ . 

Note.  The  cost  of  grading  and  paving  streets  is  sometimes  paid  in 
full  by  the  city  or  town;  sometimes  assessed  one-half  on  abutting  property, 
according  to  frontage,  sometimes  one-half  according  to  area,  sometimes  part 
by  frontage  and  part  by  area.  Owing  to  the  fact  that  comer  lots  may  be 
assessed  for  improvements  on  two  streets,  their  rates  are  sometimes  reduced. 
Areas  are  measured  from  established  building  lines. 

3.  Rockford  Avenue  (see  page  276)  is  paved  with  asphalt  from  First  to 
Second  street  at  a  cost  of  $2.25  per  square  yard.  The  area  paved  is  60 
ft.  by  360  ft.  If  one-half  of  the  cost  is  assessed  on  property  owners,  what 
sum  must  be  met  by  the  owners  of  frontage  on  one  side  of  the  street? 
If  this  cost  is  shared  according  to  frontage,  what  hts  must  be  assessed? 

Andrews,  who  owns  lots  No.  266  and  No.  265,  pays  $?;  Chapman,  for  No. 
270,  and  No.  271,  pays  $?;  Eastman,  the  owner  of  No.  267,  268,  269, 
270,  and  No.  273  pays  $?. 

4.  Find  assessments  (ex.  3)  if  comer  lots  are  assessed  on  basis  of  1/2 
frontage. 

5.  Determine  assessments  if  25%  is  levied  by  frontage  and  balance 
by  area. 

6.  A  curbing  is  laid  from  the  alley  around  these  lots,  25  ft.  from  building 
line,  at  a  cost  of  35c  per  ft.  Assess  one-half  the  cost  against  the  lots  ac- 
cording to  frontage. 

7.  The  property  fronting  on  Rockford  Avenue  is  assessed  65c  per  front 
foot  for  water  mains.     What  does  each  owner  pay? 

8.  The  alley  is  paved  at  a  cost  of  $1.20  per  sq.  yd.  from  1st  to  2nd  street. 
One-fourth  of  the  cost  is  assessed  by  area  against  lots  Nos.  264-273. 
Determine  the  "lot"  assessments. 

II.    NATIONAL  GOVERNMENT  INCOME  AND   EXPENSE. 

The  main  sources  of  National  Government  income  and 
expense  are  shown  in  the  tables  that  follow.  There  are  many 
minor  sources  also,  such  as  profits  on  coinage,  fees,  sales  of 


276 


BUSINESS   ARITHMETIC. 


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STREET 


public  land,  etc.  The  internal  tax  on  liquors,  tobacco,  etc., 
is  a  stamp  tax,  and  is  not  of  such  general  character  as  to  require 
notice.     The  other  taxes  are  paid  in  money. 


TAXATION   AND   PUBLIC   REVENUE. 


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EXERCISE. 

1.  Complete  the  tables. 

2.  Define  the  terms  in  the  first  column. 

3.  Where  does  authority  for  expenditures  rest?     What  is  an  appro' 
priation? 


278  BUSINESS  ARITHMETIC. 

Customs. 

The  National  Government  levies  duties  or  customs  on 
many  classes  of  articles,  for  revenue  and  for  protection  of 
home  industry.  Congress  has  adopted  a  tariff  or  schedule 
showing  the  rates  of  duties  on  imports;  and  a  free  list,  or 
schedule  of  articles  imported  exempt  from  duty. 

Duties  are  either  ad  'valorem  (levied  as  a  per  cent,  of  their 
net  value)  or  specific  (levied  by  measure).  Both  classes  of 
duties  are  imposed  on  some  imports.  In  general,  classes  of 
goods  varying  markedly  in  quality,  and  therefore  in  value, 
are  taxed  ''  ad  valorem.''  These  frequently  are  manufactures, 
such  as  carpets,  crockery,  knives,  etc.  Raw  materials,  and 
many  articles  of  fairly  uniform  grade,  are  taxed  specifically, 
for  example,  rivets,  gold  leaf,  eggs,  wheat. 

Ad  valorem  duties  are  not  computed  on  fractions  of  a  dollar, 
less  than  50  cents  being  discarded,  and  50  cents  or  more 
being  considered  an  additional  dollar.  The  same  rule  applies, 
in  specific  duties,  to  fractions  of  the  unit  measure.  The 
Government  employs  the  long  ton  (2240  lb.)  in  weighing, 
but  makes  allowances,  in  many  cases,  for  tear  and  breakage. 

Rates  from  a  United  States  Tariff. 

Camphor,  refined,  six  cents  per  pound. 

Castor  oil,  thirty-five  cents  per  gallon. 

Paris  green  and  London  purple,  fifteen  per  centum  ad  valorem. 

Castile  soap,  one  and  one-fourth  cents  per  pound;  medicinal  or  medicated 
soaps,  twenty  cents  per  pound;  fancy  or  perfumed  toilet  soaps,  fifty  per 
centum  ad  valorem. 

Lime,  five  cents  per  weight  of  one  hundred  pounds,  including  weight 
of  barrel  or  package. 

Cast  polished  plate  glass,  finished  and  unfinished,  or  unsilvered,  not 
exceeding  three  hundred  and  eighty -four  square  inches,  ten  cents  per  square 
foot,  above  that,  and  not  exceeding  seven  hundred  and  twenty  square 
inches  twelve  and  one-half  cents  per  square  foot;  all  above  that  twenty- 
two  and  one-half  cents  per  square  foot. 

Pen  knives,  pocket  knives,  clasp  knives,  valued  at  not  more  than  forty 
cents  a  dozen,  forty  per  centum  ad  valorem;  valued  at  more  than  forty 


TAXATION  AND   PUBLIC   REVENUE.  279 

cents  per  dozen,  and  not  exceeding  fifty  cents  per  dozen,  one  cent  per 
piece,  and  forty  per  centum  ad  valorem. 

Gold  leaf,  thirty-five  cents  per  one  hundred  leaves. 

Shingles,  fifty  cents  per  thousand. 

Horses  and  mules,  valued  at  one  hundred  and  fifty  dollars  or  less  per 
head,  thirty  dollars  per  head;  if  valued  at  over  one  hundred  and  fifty 
dollars,  twenty-five  per  centum  ad  valorem. 

Buckwheat,  fifteen  cents  per  bushel  of  forty-eight  pounds. 

Carpets,  woven  whole  for  rooms,  and  Oriental,  BerHn,  Aubusson, 
Axminster  and  similar  rugs,  ten  cents  per  square  foot,  forty  per  centum 
ad  valorem. 

Firecrackers  of  all  kinds,  eight  cents  per  pound,  the  weight  to  include 
all  coverings,  wTappers  and  packing  material. 

EXERCISE. 

1.  Define:  Income,  duties,  revenue,  tariff,  free  list,  miports,  ad  valorem, 
and  specific. 

2.  Which  is  simpler  to  levy,  a  specific  or  an  ad  valorem  tax? 

3.  State  the  kind  of  duty  specified  in  each  of  the  given  sections  of  the 
tariff.     Suggest  a  reason  for  the  kind  of  duty. 

The  country  is  divided  into  collection  districts,  each  con- 
taining a  port  of  entry  and  one  or  more  ports  of  delivery.  All 
imports  must  be  brought  first  to  the  port  of  entry  at  which  is 
situated  a  custom  house  for  the  collection  of  duties  and  for  the 
official  entry  or  clearance  of  vessels.  After  duties  are  paid 
and  permission  given  to  unload,  goods  may  be  discharged  at 
any  port  of  delivery.  Thus  the  collection  district  of  the  City 
of  New  York  has  two  ports  of  entry,  New  York  and  Jersey 
City,  which,  as  usual,  are  also  delivery  ports;  and  has  addi- 
tional ports  of  delivery  at  New  Windham,  Albany,  Cold 
Spring,  Port  Jefferson  and  Patchogue. 

In  the  larger  ports,  a  Collector  of  the  Port  is  the  chief  officer, 
having  under  him  appraisers  to  determine  values,  and  sur- 
veyors ^  weighers,  gaugers,  etc.,  to  determine  quantities. 

In  brief,  the  process  of  importing  goods  is  as  follows:  The 
seller  or  shipper  at  a  foreign  port  makes  out  an  invoice  of  the 
marks,  number,  quantities,  value,  etc.,  of  the  goods  shipped, 


280  BUSINESS   ARITHMETIC. 

weights  and  values  being  stated  in  the  standards  of  the  foreign 
country.  The  invoices,  if  over  $100,  are  certified  by  the 
local  U.  S.  consul  who  retains  a  copy  and  sends  a  second  copy 
to  the  collector  of  the  port  at  which  the  goods  are  to  be 
entered.  On  loading  the  goods  aboard  ship,  the  shipper  re- 
ceives duplicate  bills  of  lading,  or  receipts  involving  a  contract 
to  deliver  at  a  specified  port  The  shipper  forwards  copies  of 
invoice  and  of  bill  of  lading  to  the  purchaser  in  the  United 
States.  The  latter  fills  out  certain  forms  and  presents  them, 
with  the  invoice  and  bill  of  lading,  at  the  custom  house,  pays 
the  estimated  duties,  and  receives  a  permit  to  land  the  goods 
subject  to  examination  and  verification. 

The  permit  goes  to  the  Surveyor  who  is  in  direct  control  of 
the  vessel  and  its  cargo.  The  goods  are  then  landed  and 
examined  and  appraised,  to  determine  And  check  quantities 
and  dutiable  value.  One  package  in  each  invoice,  one  package 
out  of  every  ten  of  the  same  kind  and  same  price  of  each  class 
of  material,  is  examined  carefully.  The  master  of  the  vessel 
also  submits  to  the  Collector  of  the  Port  a  manifest,  or  state- 
ment of  the  name  of  the  vessel,  of  the  names  of  consignors 
and  consignees,  and  of  the  items  of  his  cargo. 

If  the  importer  does  not  care  to  use  and  pay  duty  on  his 
goods,  at  the  time  of  receipt,  he  may  give  bond  to  pay  the  duties, 
and  may  store  the  goods  in  bonded  warehouses,  maintained  by 
the  Government.  All  or  part  of  the  invoice,  within  certain 
minimum  limits,  may  be  withdrawn  as  desired  on  payment  of 
duties  and  reasonable  charges  for  storage  and  incidental  labor. 
Such  goods  may  be  withdrawn  for  export  without  being  subject 
to  duty.  If  the  goods  are  retained  in  storage  over  one  year, 
however,  the  duty  is  increased  10%.  After  three  years,  the 
goods  are  subject  to  forfeit.  If  duties  have  been  paid  and 
imports  afterwards  exported,  a  draivback  or  refunding  of  the 
duties  is  granted. 

For  the  collection  of  the  excise,  or  internal  revenue  tax,  the 


TAXATION   AND   PUBLIC   REVENUE.  281 

country  is  divided  into  districts  each  in  charge  of  a  collector. 
In  many  cases,  the  duty  is  paid  by  affixing  to  packages  of 
manufactured  goods  the  requisite  amount  of  tax  in  the  form 
of  adhesive  stamps  which  may  be  purchased  in  quantities  and 
used  as  needed.  The  stamps  must  be  destroyed  when  the 
packages  are  opened.  Drawbacks  are  allowed  on  goods  made 
in  this  country  that  are  exported  after  the  excise  tax  has  been 
paid. 

As  the  invoice  value  of  imports  is  expressed  in  the  coinage 
of  the  exporting  country,  it  is  necessary  to  reduce  such  values 
to  United  States  standards.  Coinage  reduction  tables  are 
issued  periodically  by  the  Director  of  the  United  States  Mint. 

Values  of  Foreign  Coin  in  United  States  Gold. 
(Foreign  gold  coins  unless  otherwise  stated.) 

Country.  Coin.       Abbreviation.  U  S.  Value  in  Gold 

Argentine  Republic Peso p $.965 

Austria Krone kr 203 

Belgium,  France,  Switzerland .  Franc fr • 193 

Brazil Milreis ml 546 

Chili Peso p 365 

German  Empire Mark M 238 

Great  Britain Pound  Sterling  .  £     4.8665 

Italy Lira 1 193 

Japan Yen y 498 

Mexico Peso,  Dollar  (sU.) 498 

Netherlands Florin fl 402 

Philippines Peso p 500 

Russia Ruble r 515 

Computations  under  this  table  involve  simple  multiplication. 
Illustration.     Determine  the  IT.  S.  value  of  a  shipment  of  hides  from 
Argentine,  invoiced  at  4200  pesos. 

Solution.  From  the  table,  1  peso  =  $.965.  4200  X  $-965  =  $4053,  the 
value  of  4200  pesos. 

ORAL    EXERCISE. 
.Determine  value  of  these  invoices  in  U.  S.  gold. 

1.  From  Argentine,  2000  pesos.  3.     From  France,  120Q  fr. 

2.  From  England,  £  600.  4.     From  Germany,  5000  M. 


282  BUSINESS  ARITHMETIC. 

5.  From  Chili,  250  p.  8.     From  Russia,  400  r. 

6.  From  Mexico,  1000  pesos.  9.     From  Philippines,  720  p. 

7.  From  Netherlands,  300  fl.  10.     From  Japan,  10,000  yen. 

EXERCISE. 

Determine  the  value  in  U.  S.  gold  of  these  invoices: 

1.  From  Russia,  7246.5  r.  5.     From  London,  £365.82.   ■ 

2.  From  Japan,  864.36  yen.  6.     From  Amsterdam,  4368.20  fl. 

3.  From  Switzerland,  15,329  fr         7.     From  Havre,  32,129.6  fr. 

4.  From  Germany,  2468  M.  8.     From  Naples,  673.80  1. 
9.     Of  bone  and  tallow  from  Argentine,  6754  pesos. 

10.  Of  nitrate  of  soda  from  Chili,  73,854  pesos. 

11.  Of  coffee  from  Brazil,  785.25  ml. 

12.  Of  knives  from  Sheffield,  £542.10. 

13.  Of  glass  from  Bohemia,  529.65  krone. 

Specific  Duties. 

When  specific  duties  are  charged,  the  reduction  of  foreign 
measures  to  U.  S.  standards  is  necessary.  Great  Britain  uses 
measures  similar  to  our  own,  and  most  other  exporting 
countries  use  the  metric  system. 

Once  the  invoice  quantity  is  expressed  in  U.  S.  standards, 
the  computation  of  the  specific  duty  is  a  matter  of  direct 
multiplication,  except  where  the  wording  of  the  U.  S.  tariff 
necessitates  a  further  computation. 

Illustrations.     (1)  What  is  the  duty  on  50  M.  shingles  at  50c  per  M.? 
Solution.     At  50c  per  M.  the  duty  on  50  M.  =50  X 50c  =  $25.00. 
(2)  What  is  the  duty  on  7200  lb.  buckwheat  at  15c  per  bu.  of  48  lb.? 
Solution.    The  number  of  bu.  =  7200  -^  48  =  150.     150  X  15c  =  $22.50 
duty. 

ORAL    EXERCISE. 

See  Tariff,  pp.  278,  279  for  missing  rates. 
Compute  the  duty  on: 

1.  2400  bu.  wheat  @  25c.  6.  625  lb.  camphor. 

2.  60  gr.  pens  @  12c.  7.  3000  lb.  lime. 

3.  2750  lb.  rivets  @  Uc.  8.  28  pkg.  gold  leaf. 

4.  16  T.  grindstone  @  $1.75  per  T.  9.  10  pc.  plate  glass,  24  in.X 
6.    4600  lb.  Castile  Soap.  30  in. 


TAXATION   AND   PUBLIC   REVENUE. 


283 


EXERCISE. 
Compute  the  duty  on: 

1.  246  pc.  plate  glass,  16"  X  24". 

2.  7600  ft.  bar  iron,  weighing  1.2  lb.  per  foot,  @  6/lOc  per  ft. 

3.  Three  eases  of  fire  crackers,  weighing  respectivelySlS,  129, 142  lb., 
including  all  packing  material,  @  8c  per  lb. 

4.  1520  short  tons  of  barley,  @  30c  per  bu.  of  48  lb. 

5.  320  bx.  Castile  soap,  averaging  110  lb.  per  bx.,  allowance  for  tare 


5%, 


6.  22  long  tons  of  lime. 

7.  60  gr.  bottles  of  mineral  waters  @  20c  per  doz. 

8.  Find  the  entire  duty  on  this  invoice. 

Sheffield,  Nov.  17,  191- 
Invoice  of  Hardware  Purchased. 
By  James  Evanston  of  Baltimore,  Md, 
From  The  Peter  Mfg.  Co.  of  Sheffield. 
To  he  shipped  per  Cambria. 


Marks.  Quantity.         Description. 


Price.  Amount.      Consular 
Corrections. 


^68 
<$>69 
^70 


6 

160 

42 


gr.  Files, 
lb.  Bolts, 
lb.  Wood  screws, 

Total. 

U.  S.  Money. 
Duty  on  files,  25c  doz. 
Duty  on  bolts,  l^c  lb. 
Duty  on  screws,  8c  lb. 


£ 

s 
3 
1 
1 

d 
0 
6 
9 

£ 

XX 

X 

X 

s 

X 
X 
X 

d 

X 
X 
X 

Ad  Valorem  Duties. 

The  computation  of  ad  valorem  duties  often  involves  the 
reduction  of  money  values  to  U.  S.  gold  standard,  and  always 
involves  percentage. 

Illustration.  Compute  the  duty  (40%)  on  forty  watch  cases  invoiced 
@  36  fr. 

Solution.  The  value  of  forty  watch  cases  =  40X36  fr.  =  1440  fr.  1  fr. 
=  $.193.  1440  fr.  =  1440 X.  193  =  $277.92.  The  dutiable  value  is  $278. 
The  duty  =  40%  of  $278  =  $111.20. 


284 


BUSINESS   ARITHMETIC. 


EXERCISE. 

Compute  the  duty  on: 

1.  An  invoice  of  London  purple,  valued  at  £46  8s. 

2.  8  doz.  pen  knives  @  0/1/6  per  doz. 

,      3.     40  head  of  horses  from  Canada,  @  $135. 

4.  An  invoice  of  crockery,  675  fr.    Duty  60%. 

5.  A  collection  of  orchids  from  Brazil,  @  526  ml.     Duty  25%. 

6.  An  invoice  of  swords,  valued  at  $4264.14.     Duty  50%. 

7.  A  set  of  "standard  authors"  valued  at  £64.     Duty  25%. 

8.  Compute  ad  valorem  duty  at  60%  on  this  invoice: 

Mme.  S.  Ren  ant. 
Robes  Manteaux  Modes.  62.  Place  du  TrocaMro 

Madame  C.  Fanning,  Dr. 

Paris,  le  18  Novembre,  191—  Fr. 


2 

Robes,  garni  motif 

775 

Robe,  en  foulard 

310 

Robe,  en  draps  bronze,  Princesse 

400 

Robe,  drap  gris,  brader 

526 

Blouse,  en  crepe  de  chene  blanc 

92 

Blouse,  taffeta  bleu 

65 

Note.     Both  ad  valorem  and  specific  duties  are  levied  on  many  imports. 
9.     A  rug,  6'  X  8',  valued  at  146  fr.  (see  tariff). 

10.  86  lb.  celluloid  manufactures,  valued  at  756  fr.     Duty  65c  per  lb. 
and  30%. 

11.  1200  m.  oil  cloth,  2  yd.  wide,  @  2.4  M.  per  m.     Duty  6c  sq.  yd. 
and  15%. 

12.  Extend  this  form: 

Manifest  No.  868.    Invoiced  at  Amsterdam,  Netherlands,  Nov.  29,  19 — . 

Inward  Foreign  Entry  of  Merchandise. 
Imported  by  J.  J.  Jervis  In  the  steamer  Kriedam 

A.  B.  Roe  Master  From  Amsterdam.    Arrived  Dec.  15,  19 — 


No. 


920-1 
922-3 


Packages  and  contents 


2  cas.  Brussels  carpet 

3/4  yd.  wide. 
2   cas.  Tap.  3/4  yd. 

wide. 

40%  of  $ 

?  yd.  @  44c 
?  yd.  @  40c 


Quan- 
tity 

Free 
List 

44^8q.yd. 
and  40^  ad 
valorem 

40^Bq.  yd. 
and  405t  ad 
Talorem 

Duty 

400m. 
420m. 

600  fl. 

$? 

840  fl. 

$? 

$? 
? 
? 

Total 


CHAPTER   XXXV. 

INTEREST. 

Interest  ranks  with  the  fundamental  processes,  and  with 
simple  percentage,  as  one  of  the  most  useful  branches  of 
arithmetic.  It  has  more  general  and  varied  applications  than 
simple  percentage,  and  its  use  is  being  constantly  extended. 
It  is  the  great  interpretative  branch  of  arithmetic  and  has  to 
do,  especially,  with  the  study  of  values. 

INTRODUCTORY    EXERCISE. 

1.  Give  business  reasons  for  borrowing  money. 

2.  Need  a  prosperous  business  ever  borrow  money? 

3.  Why  is  money  paid  for  the  use  of  others'  money? 

4.  I  owe  Smithson  $500,  due  to-day.  If  I  delay  payment  for  a  month, 
does  Smithson  gain  or  lose?    Why? 

5.  Does  the  time  I  keep  another's  money  affect  the  charge  for  its  use? 

Interest  has  double  meaning.  Strictly,  it  is  the  use  of  money. 
Commonly,  it  is  the  sum  paid  for  that  use.  The  sum  on  which 
interest  is  charged  is  the  principal.  Interest  is  computed  as 
a  per  cent,  of  this  principal  per  year,  with  relatively  less 
amounts  for  portions  of  a  year.  Each  state  has  a  legal  rate 
of  interest,  used  when  no  rate  is  stated  in  interest-bearing 
obligations.  By  agreement,  a  lower  rate  may  be  charged;  or 
in  some  states,  a  higher  rate  up  to  a  maximum  fixed  by  law 
may  be  charged.  A  requirement  to  pay  higher  than  the  legal 
maximum  is  termed  usury. 

Commercially,  the  interest  year  is  reckoned  as  360  days, 
or  12  months  of  80  days  each.  If,  however,  the  time  is  a  year 
or  more,  calendar  years  are  taken  for  the  full  years.  The 
United  States  Government,  and  some  banks,  reckon  interest 

285 


286  BUSINESS  ARITHMETIC. 

on  a  basis  of  365  or  366  days.  Interest  tables  commonly  are 
used  to  save  computation.  For  long  periods,  interest  time  is 
computed  by  compound  subtraction;  for  short  periods  the  exact 
number  of  days  is  used,  as  a  rule,  this  depending,  however,  on 
the  custom  of  the  business  or  the  character  of  the  obligation. 

ORAL    EXERCISE. 

1.  Determine  the  interest  on  $6000  for  one  year  at  6%.     At  8%. 

2.  Find  the  interest  on  $1200  for  1  year  at  5% ;  for  2  years ;  for  6  months; 
for  2  months. 

3.  What  part  of  the  interest  for  a  year  is  the  interest  for  3  mo.?  2  mo., 
4  mo.,  60  da.,  90  da.,  80  da.,  120  da  ,  30  da.? 

4.  Determine  the  interest  on  $800,  at  4%,  for  1  yr.,  3  mo.,  4  yr.,  2  mo., 
90  da.,  6  mo.,  30  da.  For  a  fixed  rate  and  principal,  how  is  interest  affected 
by  a  change  of  time? 

5.  How  does  interest  at  4%  compare  with  interest  at  5%,  on  the  same 
principal? 

6.  If  the  interest  for  a  fixed  time,  on  a  given  principal  is  $32,  at  4%, 
find  the  interest  at  5%,  at  6%,  2%,  8%,  1/2%,  4|%.  What  is  the 
effect  of  increasing  or  decreasing  the  rate,  if  principal  and  time  remain 
constant? 

7.  The  interest  on  a  given  principal,  at  a  fixed  rate,  for  20  days,  is 
$1.60.  Find  the  interest  for  5  da.,  10  da.,  30  da.,  40  da.,  2  mo.,  80  da.,  11  da., 
3  mo.     What  is  the  effect  of  change  of  time? 

8.  If  rate  and  time  remain  unchanged,  how  will  the  increase  or  decrease 
of  principal  affect  the  interest?  If  the  interest  on  $1200,  for  a  certain 
time,  is  $36,  determine  the  interest  on  $900,  $800,  $700,  $6000,  $720,  $400. 

Most  practical  methods  of  computing  interest  depend  on  the 
principles  just  emphasized,  are  closely  related,  and  are  based 
on  a  year  of  360  days.  It  is  best  to  understand  several 
methods  and  to  select  for  general  use  the  one  that  seems  most 
natural  and  simple. 

I.    CANCELLATION   METHOD. 

(Based  on  360  or  365  days  to  the  year.) 

Illustration.  Example.  Compute  the  interest  on  $720  for  84  daya 
at  6%. 


INTEREST.  287 

Analysis  and  Solution.     The  interest  for  1  year  is  6/100  of  $720.     Fol 
84  davs,  it  is  84/360  of  this  amount. 

2 

4720rX. 06X84 


■360- 
1 


$2  X. 06X84  =  $10.08. 


EXERCISE. 

Compute  the  simple  interest  on: 

1.  $720  for  36  da.  at  8%.  5.     $1260  for  0  mo.  15  da.  at  6%. 

2.  $925  for  2  mo.  6  da.  at  6%.  6.     $829.60  for  39  da.  at  4%. 

3.  $456.50  for  8  mo.  at  7^%.  7.     $3600  for  20  da.  at  5%. 

4.  $862.85  for  7  mo.  3  da.  at  6%.        8.     $240  for  18  da.  at  9%. 
9.     $1200  from  Jan.  5  to  Nov.  21  at  6%.     (Approximate  time.) 

10.  $12,900  from  Aug.  21  to  Dec.  3,  at  4%.     (Approximate  time.) 

11.  $270  from  Mar.  13  to  May  24  at  7%.     (Exact  time.) 

12.  $1200  for  75  da.  at  5% 

13.  $320  for  3  mo.  5  da.  at  8%. 

II.    SIXTY-DAY   METHOD. 
The  sixty-day,  or  bankers'  method,  is  one  of  the  most  effec- 
tive in  use.     By  it,  interest  is  first  computed  at  6%  and  then 
reduced  to  the  required  rate. 

INTRODUCTORY    EXERCISE. 

1.  If  the  interest  rate  is  6%,  what  per  cent,  of  the  principal  is  the 
interest  for  6  mo.?  2  mo.?  8  mo.?  1  mo.? 

2.  What  per  cent,  for  60  da.,  30  da.,  90  da.,  20  da.? 

3.  For  what  period,  in  months,  is  the  interest  1%  of  the  principal? 
For  what  period  in  days? 

4.  What  is  the  simplest  method  of  finding  1%  of  a  number? 

5.  Find  the  interest  at  6%,  for  60  days,  on  $1200,  $800,  $300. 
Illustrative  Example.     Compute  the  interest  on  $7200  for  96  days 

at  6%.     At  3%. 

Solution.     The  interest  for  60  days  at  6%  equals  1%  of  the  principal. 
$7200.00  =  the  principal. 

72.00  =  interest  for  60  da.  (1%  of  principal) 

36.00=      "        "  30   "    (30  da.  =  1/2  of  60  da.     .'.  interest  equals  1/2 

of  interest  for  60  da.) 
7.20=      "         "    6_  "    (6  da.  =  1/10  of  60  da.) 
$115.20=       "         "   96   "    Ans.  (6%) 
The  mterest  at  3%  =  ^he  interest  at  6%  or  $57.60. 


288 


BUSINESS   ARITHMETIC. 


Example  2      Compute  the  interest  on  $840  for  4  mo.  24  da.  at  6%. 
Solution. 

$840      =the  principal. 

8  40  =  interest  for    2  mo.  (Use  a  line  to  mark  decimal  point.) 

8  40=       " 

2  80=       "         "   20  da.    (20  da.  =  1/3  of  2  mo.) 

56=       "         "  ^^(  4  da.  =  1/5  of  20  da.) 

"     4  mo.  24  da. 


$20 


16  = 


$  8 

40  = 

16 

80  = 

2 

10  = 

84  = 

42  = 

$20 

16  = 

Check.     Examples  may  be  checked  by  changing  time  components. 
Illustrations.     (Checks  on  example  2  ) 
4  mo.  24  da.  =  144  da. 
(1) 

interest  for    60  da. 

"  120  da.  (Twice  interest  for  60  da.) 

"  15    "    (1/4  mterest  for  60  da.) 

"  6   "    (1/10      "        ''   60  ."  ) 

"  3   ''    (i           "        "  6  da.  ) 

(2) 
$  8 140  =  interest  for   60  da. 

21100=      "        "   150   "    (2f  times  interest  for  60  da.) 
|84=      "        "       6   " 
$20116=      "         "    144    "~  (By  subtraction.) 
Note.     This  method  substitutes  simple  addition  and  division  for  more 
complicated  operations,  and  offers  positive  check  solutions.     If  times  are 
large,  600  days  may  be  used  as  a  unit  time  in  place  of  60  days;  if  the  time 
is  small,  three  places  may  be  pointed  off  in  order  to  find  interest  for  a  unit 
time  of  6  days. 

It  is  generally  advisable  to  select  such  sub-times  that  the  interest  at  each 
step  may  be  found  from  the  previous  step,  or  from  the  unit.  See  ex.  1, 
above. 

ORAL    EXERCISE. 

Subdivide  these  times  into  parts,  and  also  suggest  a  "check"  series: 

1.  84  da.                       5.     5  mo.  7  da.  9.    3  mo.  21  da. 

2.  90  da.                        6.     87  da.  10.     48  da. 

3.  11  da.                       7.    4  mo.  15  da.  11.    37  da. 

4.  21  da.                       8.    7  mo.    3  da.  12.     16  da. 


ORAL    EXERCISE. 

Compute  the  interest  at  6%  on: 

1.  $1200  for  60  da. 

2.  $  840  for  30  da. 


$  960  for  6    da. 
$  120  for  20  da. 


5.    $  600  for    3  da. 

9.     $  884  for  30  da. 

6.     $640  for  45  da. 

10.     $  390  for    2  da. 

7.     $  900  for  2  mo.  6  da. 

11.     $  900,  2  mo.  20  da. 

8.     $  126  for  600  da. 

12.    $1600  for  15  da. 

Compute  the  interest  on: 

13.     $300,  4  mo.,  5%. 

15.     $800,  45  da.,  4^%. 

14.    $240,  30  da.,  7%. 

EXERCISE. 

Compute  the  interest  on  the  following,  checking  by  change  of  time: 
Principal.     Time.     Rate.  Principal.         Time.  Rate. 

1.  $  542.50     17  da.     6%.  5.     $2164  23  da.     6%. 

2.  $3690         97  da.     5%.  6.    $  980         3  mo.  8    da.     6%. 

3.  $  452.80    48  da.     4%.  7.     $  452.75     5  mo.    6  da.     6%. 

4.  $  725         39  da.     6%.  8.    $3726.        6  mo.  13  da.     4%. 
Compute  the  amount,  principal  plus  interest,  on  the  following: 

9.     $426     11  da.     6%.  13.    $1268  132  da.     5%. 

10.  $  300    25  da.     8%.  14.    $  282  90    3  mo.    17  da.     6%. 

11.  $1285     34  da.     6%.  15.    $  562.80     1  mo.      5  da.     6%. 

12.  $  950    68  da.     6%.  16.    $1262.50    7  mo.    15  da.     7§%. 
Using  approximate  time,  compute  the  interest  on: 

17.  $9650  from  Jan.  27  to  Oct.  19,  at  6%. 

18.  $248.20  from  Nov.  12  to  Jan.  13,  at  8%. 

19.  $3242.90  from  Mar.  28  to  Dec.  19,  at  6%. 
Using  exact  time,  compute  the  amount  of: 

21.  $1220  from  Mar.  22  to  Apr.  14,  at  6%. 

22.  $356  from  Feb.  12,  1909,  to  June  14,  1909,  at  7%. 

In  order  to  simplify  computation,  the  numerical  value  of 
the  principal  and  time  (in  days)  may  be  interchanged  without 
affecting  the  resulting  interest.  In  finding  the  amount, 
however,  it  is  necessary  to  remember  to  add  the  original 
principal. 

Illustration.  The  interest  on  $120  for  17  days  is  the  same  as  that 
of  $17  for  120  days.     Thus  in  the  first  case,  the  factors  are 

120  X 17  X. 06, 
360 


290         ^  BUSINESS   ARITHMETIC. 

and  in  the  second  case,  they  are  simply  interchanged  in  order, 

17  X 120  X  .06. 
360 
Example.     Compute  the  interest  on  $72  for  185  da. 
Solution.    Read  the  example:  Compute  the  interest  on  $185  for  72  da. 
$1.85  =  interest  for  60  da. 
.37=       "         "    12   " 
$2  22=      "        "  72   "    on  $185,  or  for  185  da.  on  $72. 

ORAL    EXERCISE. 

By  interchanging  terms,  compute  interest  at  6%,  on: 

1.  $60  for  29  da.  4.    $120  for  38  da.  7.    $600  for  71  da. 

2.  $20  for  96  da.  5.     $15  for  92  da.  8.     $150  for  42  da. 

3.  $30  for  47  da.  6.     $12  for  132  da. 

EXERCISE. 

Compute  interest  and  amount  on: 

1.  $126  for  197  da.  at  6%.  3.    $3000  for  179  da.  at  6%. 

2.  $540  for  83  da.  at  8%.  4.    $  450  for  168  da.  at  6%. 

III.    SIX  PER  CENT.   METHOD. 
In  case  the  interest  term  is  greater  than  one  year,  the  6% 
method  is  frequently  used.     Some  prefer  it,  also,  for  short 
terms. 

INTRODUCTORY    EXERCISE. 

1.  The  interest  on  $1.00  for  one  year  at  6%  =  ? 

2.  The  interest  on  $1.00  for  2  mo.  =  ?  .  Find  the  interest  for  30  da., 
6  mo.,  4  mb.  What  is  the  interest  on  $1.00  for  any  multiple  of  2  mo.  or 
60  da.? 

3.  The  interest  on  $1.00  for  6  da.  =  ?  .  How  may  it  be  found  for  2  da., 
6  da.,  for  any  multiple  of  6  da.? 

4.  The  interest  on  $1 .00  for  4  mo.  =  ? .  From  it,  determine  the  interest 
on  $400  for  the  same  time. 

Theoretically,  the  6%  method  consists  in  applying  the 
60-day  method  to  building  up  the  interest  on  $1.00,  multiply- 
ing the  result  by  the  number  of  dollars.  For  practical  use, 
the  following  values  should  be  kept  in  mind : 


INTEREST. 


291 


$.06      =  interest  on  $1.00  for  1  yr.    at  6%. 

.01  =  "  "  "  "  2  mo.  "  " 
or    .005    =        "        "      "       "    1    "     "     " 

.001  =  "  "  "  "  6  da.  "  " 
or    .000J=        "        "      "       "    1    "      "     " 

Illustration.  Example.  Compute  the  interest  at  6%  on  $700, 
for  1  yr.  7  mo.  18  da. 

Solution.    $.06   =  interest  on  $1.00  for    1  yr. 

.035=      "        "      "       "     7mo.  (S^Xint.  for2mo.). 
■003=       "        "       "       "    18da.  (3Xint.  for6da.). 
$.098=       "        "      "       "      1  yr.  7  mo.  18  da. 
700  X $.098  =  interest  on  $700  for  given  time  =  $68.60. 

Note.  Interest  at  other  rates  may  be  obtained  by  ratio,  applied  either 
to  the  interest  on  $1.00,  or  to  the  .total  interest.  Thus  2/3  of  either  $.098, 
or  of  the  product,  $68.60,  equals  the  interest  at  4%. 


ORAL    EXERCISE 

Find  the  interest  on  $1.00,  at  6%,  for: 

1.  88  da.  5.    29  da. 

2.  92  da.  6.    62  da. 

3.  17  da.  7.    54  da. 

4.  14  da.  8.      2  da. 
Compute  the  interest  on: 

13.  $200  for  24  da. 

14.  ♦  $1000  for  3  mo.  6  da. 


9. 
10. 


1  mo.    9  da. 

2  mo.  27  da. 


11.  3  mo.  14  da. 

12.  1  yr.  4  mo.  11  da. 

15.  $500  for  2  yr.  6  mo.  12  da. 

16.  $400  for  96  da. 


EXERCISE. 


Compute  the  interest  on: 

Principal, 

Time. 

Rate. 

1.     $60432 

1  yr.  2  mo. 

8  da. 

6%. 

2.    $      98.25 

2  yr.  5  mo. 

3  da. 

6%. 

3.    $  7294 

3  yr.  1  mo. 

15  da. 

6%. 

4.    $     520 

8  mo. 

18  da. 

3%. 

5.    $  2460 

lyr. 

16  da. 

7%. 

6.    $    590.84 

1  yr.  4  mo. 

12  da. 

5%. 

7.    $  1580 

3  mo. 

16  da. 

5%. 

8.     $  2050.40 

5  mo. 

13  da. 

6%. 

292  BUSINESS  ARITHMETIC. 

Principal.  Time.  Rate. 

9.    $    840  7  mo.  24  da.  G%. 

10.  $  1292  2  yr.  5  mo.  20  da.  6%. 

11.  $36,432,  from  Jan.  29,  1908,  to  Oct.  17,  1909,  at  6%. 

12-21.     Solve  by  6%  method,  examples  1  to  8  and  21-22,  page  (289). 
Find  the  interest  on  the  following  by  the  6%  method  and  "check"  by 
the  60-day  method: 

22.  $324  for  1  yr.  2  mo.  8  da.  at  6%. 

23.  $98.40  for  2  yr.  6  mo.  15  da.  at  8%. 

24.  $7250  for  3  mo.  21  da.  at  4  %. 

Find  the  amount  of  the  following,  using  60-day  method  and  checking 
by  the  6%  method: 

25.  $76.80  from  Jan.  12  to  Aug..  17,  at  6%. 

26.  $358.25  from  Feb.  14  to  Dec.  29  at  6%. 

IV.    TABLE  METHOD. 
Interest  tables  of  many  forms  are  in  common  use  among 
bankers  and  brokers.     They  usually  require  decimal  division 
of   the   principal   and    ordinary   subdivision   of   time.     The 
following  outline  illustrates  one  common  form. 

Six  Per  cent.  Interest  Table. 

Time.        $10       $20       $30       $40       $50       $60       $70       $80       $90 

1  da.     .00167  .00333  .00500  .00667  .08333  .01000  .01167  .01333  .01500 

2  da. 

3  da. 

1  mo. 

2  mo. 

Note.  The  practical  tables  give  interest  for  each  day  from  1  to  30. 
Such  tables  should  be  constructed  with  the  utmost  care  and  should  read 
to  five  or  more  decimal  points,  to  avoid  errors  when  applied  to  large 
amounts. 

Illustration  op  Error.  Suppose  an  interest  table  is  constructed  to 
show  interest  at  6%  for  twenty  days,  on  amounts  of  $1.00  to  $99.00.  In 
a  three-place  table  the  interest  for  $10.00  would  be  $.033;  in  a  seven-place 
table,  $.0333333.  Applying  the  table  to  determine  interest  on  $100,000, 
it  is  necessary  to  move  the  decimal  four  places  to  the  right.  By  the  small 
table  this  would  give  $330;  by  the  seven  place,  $333.33 — a  diflference  of  $3.33. 


INTEREST.  293 

EXERCISE. 

1.  Complete  the  table  (p.  292)  by  computation,  and  from  known  values. 

2.  Show  how,  from  the  given  times,  to  obtain  the  interest  for  any- 
time. Find  the  interest  on  $10  for  17  da.  (Suggestion,  17  da.  =  10  da. 
plus  7  da.),  23  da.,  4  mo.  6  da. 

3.  Show  how  to  use  the  table  in  finding  interest  on  any  principal, 
(a)  if  a  decimal  of  a  given  principal;  (6)  if  an  irregular  amount.  Find  the 
mterest  on  $40,000  for  1  da.;  on  $43  for  1  da.  (Suggestion.  $43  =  $40+ 
$3.) 

4.  Using  the  table  just  constructed,  determine  the  interest  in  ex. 
1-11  of  the  preceding  exercise. 

Exact  Interest. 
Exact  interest,  based  on  365  days,  may  be  computed  by  the 
cancellation  method,  or  it  may  be  obtained  from  interest 
computed  on  a  360-day  basis.  Thus  1  day's  interest,  on  a 
business  basis,  is  1/360  of  a  year's  interest;  on  an  exact  basis, 
it  is  1/365.  The  difference  is  1/360  -  1/365,  or  1/73.  That 
is,  exact  interest  is  1/73  less  than  common  interest. 

Illustration.  Example.  Compute  exact  interest  on  $450  for  84  da., 
at  6%. 

Solution.    $4.50  =  interest  for  60  da. 
1.50  =       "        "  20   " 
.30  =       "        "  _4   " 
$6.30  =       "         "   84   "         " 
,  $6.30  ^  73  =  $.09. 

$6.30  -  $.09  =  $6.21,  the  exact  interest. 

EXERCISE. 
Find  the  exact  interest  on: 

1.  $582.90  for  39  da.,  at  6%.  5.     $2960  from  Aug.  17  to  Dec.  29, 

2.  $1254  for  87  da.,  at  5%.  at  6%. 

3.  $632.96  for  215  da.  at  6%.         6.    $382.50  from  Jan.  5  to  June  26 

4.  $98.25  for  49  da.,  at  4%.  at  6%. 

Interest  Problems. 

Interest  is  a  truer  measure  of  profit  and  loss  than  simple 
percentage,  and  is  practically  the  common  instrument  for 


294  BUSINESS  ARITHMETIC. 

the  computations  incident  to  financial  operations.  The 
problems  that  follow  are  selected  at  random  to  show  common 
uses  of  interest. 

EXERCISE. 

1.  I  lend  James  Brown  $2400  on  Jan.  6,  1909,  which  he  repays  with 
interest  on  Aug.  15,  1910.  What  does  he  pay  in  settlement?  What  does 
he  pay  for  the  use  of  the  money? 

2.  I  borrow  $1365,  on  my  written  promise  to  pay  in  120  days  with 
5%  interest.     At  the  end  of  the  period,  what  is  due  in  settlement? 

3.  What  is  the  quarterly  income  from  a  $20,000  loan  placed  at  4|% 
interest? 

4.  $824,  if  placed  in  a  savings  bank  paying  3|%  interest,  will 
amount  to  what  sum  in  6  months? 

5.  If  money  is  worth  8%  to  a  buyer,  is  it  better  for  him  to  pay  $880 
cash,  or  to  wait  3  months  and  pay  $900? 

6.  Determine  the  per  cent,  of  gain  on  a  four  months'  investment  of 
$1280,  which  yields  5%  interest. 

7.  If  goods  are  bought  for  $420,  is  it  better  to  sell  them  in  three  months 
at  5%  profit,  or  at  5%  interest  on  cost? 

8.  I  am  thinking  of  buying  a  piece  of  property  for  $8750  which,  it  is 
claimed,  will  rent  for  enough  to  yield  9%  net  interest  on  investment.  What 
sum  must  be  cleared?  If  repairs,  taxes,  etc.,  are  estimated  at  $240  per 
year,  what  monthly  rental  should  be  charged? 

9.  A  bill  of  crockery,  $752.80,  was  bought  Jan.  27,  on  2  months  credit. 
It  was  not  paid  until  June  17.     What  was  then  due?     Interest,  6%. 

10.  I  lent  through  a  broker,  for  a  term  of  6  months :  $500  at  5  %  interest ; 
$800  at  4:1%;  $1200  at  5h%.  The  broker  charged  1/4%  interest  as  his 
commission.     Determine  my  net  income. 

11.  If  this  account  draws  6%  interest,  what  is  due  on  Aug.  10  of  the 
year  following? 

S.  R.  Ramey 


191— 

Jan.    17 

300.— 

Mar   15 

286.— 

July    12 

125.— 

Aug.  10 

85.60 

12.        The  Montrose  County  School  Fund  is  invested  in: 
$10,000  City  of  Minneapolis  3^%  bonds. 
$40,000  Morristown  Water  Works  bonds,  bearing  4i%. 
$120,000  Lewiston  Gas  Co.  bonds,  bearing  5%. 
What  is  the  annual  income  from  the  fund? 


INTEREST.  295 

13.  I  owe  a  London  exporter  £266  5  s.  10  d.,  with  interest  at  6%* 
from  Mar.  12  to  Jan.  8.     What  is  due? 

CREDIT  PRICES. 
ORAL    EXERCISE. 

1.  Is  it  better  for  a  merchant  to  sell  an  article  for  $400  cash,  or  for 
$400  credit? 

2.  If  he  gives  credit  what  does  he  lose? 

3.  Should  a  credit  price  be  greater,  or  less,  than  a  cash  price? 

4.  If  the  worth  of  money  to  a  merchant  be  known,  how  may  a  credit 
price,  equivalent  to  a  cash  price,  be  determined? 

Note.  While  many  merchants  add  arbitrary  amounts  to  the  cash 
price  when  selling  on  credit,  others  use  a  carefully  computed  rate  of  interest 
in  determining  credit  prices. 

Determine  credit  prices  for  the  following: 
Cash  Price.  Credit  Term.  Money  Worth. 

5.  $  8.40  2  mo.  6%. 

6.  80.00  1  mo.  7i%. 

7.  6.50  30  da.  12%. 

8.  5.00  3  mo.  8%. 

9.  240.00  75  da.  9%. 

EXERCISE. 

(Use  6%  as  a  rate  unless  otherwise  stated.) 

1.  Does  it  pay  to  buy  flour  on  5  months  credit,  at  $6.50,  if  it  can  be 
bought  for  $6.00  cash,  and  money  is  worth  8%? 

2.  What  profit  results  from  buying  a  $16.50  article  on  5  months 
credit,  and  immediately  selUng  it  at  the  same  price  on  1  month  credit? 

3.  What  is  a  fair  settlement  for  a  bill  of  merchandise  amoimting  to 
$1240  paid  3  months  before  its  4  months  credit  term  is  up? 

Note.  Subtract  interest  for  period  before  due.  This  is  the  interest- 
discount  method. 

TO  DETERMINE  RATE  OF  INTEREST. 
INTRODUCTORY    EXERCISE. 

1.  If  the  interest  on  $800  for  1  year  is  $36,  what  is  the  rate  of  interest? 

2.  If  the  interest  on  $600  for  3  months  is  $15,  what  is  the  interest  for 
a  year.     The  year's  interest  is  what  per  cent,  of  the  principal? 


296  BUSINESS  ARITHMETIC. 

3.  What  is  the  interest  on  $1200  for  9  days  at  6%?  At  1%?  $7.20 
is  the  interest  at  what  rate  for  the  same  time? 

It  is  evident  that  if  the  interest  for  a  year,  on  any  principal, 
is  known,  or  can  easily  be  obtained,  the  rate  of  interest  is 
determined  by  dividing  the  known  interest  by  1%  of  the 
principal.  If  the  time  is  irregular,  it  is  usually  easier  to 
measure  the  given  interest  by  the  interest  on  the  given  prin- 
cipal, at  1%  for  the  given  time. 

Illustration.     At  what  rate  does  $720  earn  $4  in  40  days? 
Solution  (1). 

$  4.00  =  interest  for    40  days. 
36.00  =       "        "  360     "     (1  yr.) 
$36.00  -r-  $720  =  .05. 
The  rate  is  5%. 
Solution  (2).     The  principal  is  $720. 
$7.20  =  interest  at  6%  for  60  da. 
4.80  =        "        "  6%   "   40  " 
.80  =        "         "  1%    "   40  " 
$4  (given  interest)  -i-  $.80  (interest  at  1%)  =  5. 
The  rate  is  5%. 

ORAL    EXERCISE. 

Principal  and  time  being  constant. 

1.  If  the  interest  at  6%  is  $48,  $60  is  the  mterest  at  ?  %. 

2.  $3.60  is  the  interest  at  ?  %,  if  $9  is  interest  at  2\%. 

3.  84c  is  the  interest  at  ?  %  if  16  c  is  the  interest  at  2%. 

Find  the  rate  of  interest  if 

4.  $1200  earns  $72  in  one  year.  6.     $400  earns  $3  in  2  months. 

5.  $800  earns  $20  in  6  months..  7.    $600  earns  36c  in  12  days. 

EXERCISE. 

1.  If  the  Russian  Government  sells  its  last  6%  bond  issue  at  10% 
below  face,  what  rate  of  interest  does  it  really  pay  on  the  funds  that  it 
secures? 

2.  Discuss  the  investment  value  of  a  6%  investment  running  one  year 
with  a  5%  one  running  ten  years. 

3.  The  offer  of  $80  for  the  use  of  $1500  for  45  days  is  equivalent  to  an 
offer  of  ?  %  interest. 


INTEREST.  297 

4.  $800  is  loaned  from  January  15  to  September  12  at  6%,  and  from 
October  1  to  January  1,  at  5%.     What  annual  rate  is  earned? 

PROFIT  AND  LOSS   MEASURED  BY  INTEREST. 
INTRODUCTORY    EXERCISE. 

1.  Is  it  better  to  allow  3  months  or  6  months  credit  on  merchandise 
sold  at  a  fixed  price? 

2.  Is  it  better  to  earn  5%  profit,  or  5%  interest,  on  an  article  that  is 
held  2  years  before  sale?     On  one  held  6  months? 

3.  When  does  a  rate  of  interest  yield  a  higher  profit  than  the  same  rate 
of  gain? 

4.  Which  more  clearly  shows  the  real  return  from  an  investment — the 
rate  of  interest,  or  the  rate  of  gain?    Why? 

The  quickness  of  sale,  after  purchase,  greatly  affects  the 
profits  of  a  business.  This  is  the  period  of  "turn-over  of 
capital."  The  above  exercise  emphasizes,  as  in  percentage, 
the  value  of  a  rapid  turn-over,  and  shows  the  mathematical 
basis  for  the  old  business  motto  "Quick  sales  and  small 
profits." 

EXERCISE. 

1.  A  merchant  invests  in  articles  costing  $2.40,  which  sell,  on  an 
average,  in  2  mo.  6  da.,  at  3%  profit.  What  is  his  annual  rate  of  income 
on  investment? 

2.  Property  bought  January  29,  for  $5000,  and  sold  June  26  for  $5400, 
yields  what  rates  of  gain  and  interest? 

3.  A  house,  bought  for  $9600,  is  rented  for  $50  per  month,  except  for 
2  months  of  the  year.  Taxes  are  lj%  on  an  assessed  value  of  $8000, 
and  repairs  cost  $118.60.     What  is  the  per  cent,  of  income  on  investment? 

4.  Is  it  better  to  make  5%  interest  or  3%  profit  on  a  7  months  in- 
vestment.    Illustrate. 

5.  A  real  estate  dealer  advertises  as  a  10%  investment  for  $5000,  a 
house  that  rents  for  $50  per  month.  Taxes  are  1^%  on  an  assessed 
value  of  $4200;  msurance  is  11%  on  $3000.  Is  allowance  made  for 
repairs? 


298  BUSINESS  ARITHMETIC. 

TO  FIND  THE  PRINCIPAL. 
INTRODUCTORY    EXERCISE. 

1.  What  is  the  interest  on  $1  at  6%  for  2  months?  $4  is  the  interest 
on  what  principal  for  the  same  rate  and  time? 

2.  What  is  the  amount  of  $1  for  6  months  at  10%.  $525  is  the  amount 
of  how  many  dollars,  same  time  and  rate? 

3.  What  is  the  measure  and  quantity  measured  in  examples  1  and  2  ? 

.      -ri     1  .      T»  •     •     1    Total  interest  ,.  ^.  ,     ,  v 

4.  Explam:  Prmcipai  =  :j— ^  (for  same  tmie  and  rate). 

Interest  on  3pi 

Illustration.     (1)  What  principal  earns  $18.30  in  12  days  at  6%? 
Solution. 

Int.  on  $1  for  12  da.  is  $.002.  If  $1  under  given  conditions, 

$18.30  ^  $.002  =  $9150.  earns  $.002,  $18.30  represents  the 

The  principal  is  $9150.  earning    of    as    many    dollars    as 

$.002  is  contained  times  in  that 
amount. 
Illustration  (2).     $976.32    is   the   amount    of   what   principal   for 
3  mo.  12  da.  at  6%? 
Solution. 

The  amount  of  $1  for  3  mo.  12  da.  at  6%  is  $1,017.^ 
$976.32  -^  $1,017  =  $960.     The  principal  is  $960. 

ORAL    EXERCISE. 

Find  the  principal. 

Int.  Time.        Rate. 

1.  $30.         18  da.        6%. 

2.  $48.  6  mo.       8%. 

3.  $  9.60     36  da.        6%. 

7.  What  principal  amounts  to  $550  in  2  years  at  5%? 

8.  What  principal  amounts  to  $2080  in  6  mo.  at  8%? 

EXERCISE. 

1.  On  January  1,  1911,  a  corporation  must  meet  a  debt  of  $25,000. 
What  sum  set  aside  on  October  15,  1909,  at  4%  will  amoimt  to  the  debt 
when  due? 

2.  What  sum  must  be  loaned  at  b\%  to  earn  a  quarterly  income  of 
$180? 

3.  Capitalize  at  6%,  semi-annual  earnings  of  $84,200. 

Note  To  capitalize  earnings  or  expense  is  to  determine  the  capital, 
or  principal  that,  at  the  given  rate,  will  produce  the  stated  earnings. 


Int. 

Time. 

Rate. 

4.    $3.20 

24  da. 

6%. 

5.    $9. 

1  yr.  6  mo. 

3%. 

6.    $2.40 

3  mo. 

8%. 

INTEREST.  299 

4.  Capitalize  at  5%  annual  charges  for  rented  buildings  for  which  a 
city  government  pays  $18,600. 

5.  By  borrowing  at  4%,  a  city  government  might  have  how  much 
money  with  which  to  secure  its  own  buildings,  for  the  $16,500  it  now  pays 
annually  in  rentals? 

The  present  ivorth,  or  true  value,  of  a  future  payment  or 
debt,  is  the  principal  that,  at  the  "  worth  of  money,"  will 
amount  to  the  payment  when  due.  Present  worth  is  some- 
times used  in  place  of  interest-discount,  in  reckoning  advance 
payments,  in  comparing  bids  and  investments. 

Illustration.     Compare  the  present  value  of  a  $960  cash  ofifer  and  a 
$964.25  credit  price  for  6  months,  money  being  worth  6%. 
Solution. 

The  amount  of  $1  for  3  months  at  6%  is  $1,015. 
$964.25  -=-  $1,015  =  950. 
The  present  value  of  the  credit  price  is  evidently  $950.     The  credit 
price  is  better  for  the  buyer  than  the  cash  price. 

Question.  Why  is  the  present  worth  method  more  accurate  than  the 
interest-discount  method? 

EXERCISE. 

1.  Money  being  worth  8%  to  me,  what  do  I  gain  by  buying  property 
for  $960,  on  6  months  credit,  and  immediately  disposing  of  it  for  cash, 
at  the  same  price? 

2.  By  present  worth,  at  6%,  compare  these  bids  for  the  construction 
of  a  building: 

C.  Smith  &  Son,  $12,000:  1/4,  cash,  1/4,  6  mo.;  balance,  1  yr. 
James  Andrews;  $11,750:  1/3,  4  mo.;  balance,  1  yr. 
A.  B.  Norton,  $11,500:  1/2,  4  mo.;  balance  1  yr. 

3.  If  money  is  worth  6%,  what  sum  fairly  settles,  on  June  15,  a 
payment  of  $1200  not  due  until  August  20? 

4.  What  sum  settles  in  cash  for  a  purchase  of  $4284,  on  terms  of  1/4 
cash,  balance  90  days.     Money  is  worth  5%. 

TO  FIND  THE  TIME. 
INTRODUCTORY    EXERCISE. 

1.  If  6c  is  the  interest  for  1  day,  on  a  certain  principal,  54c  is  the  interest 
for  how  many  days? 


300  BUSINESS  ARITHMETIC. 

2.  Find  the  interest  of  $720  for  1  day.    84c  is  the  interest  on  the  same 
principal  for  how  many  days? 

o      T7«     1  •        rr.-        -J  Total  interest  ,  .  ,      ,  x 

3.  iixplam.     Time,  m  days,  =  ,       . — fTT^  (same  prm.  and  rate). 

4.  Show  how  to  use  one  month,  or  one  year,  as  a  measure,  in  place  of 
one  day. 

Illustration.     In  what  time  does  $840  earn  $28  at  6%? 

Solution. 

The  interest  on  $840  for  1  da.  =  14c'. 

$28  -h  $.14  =  200. 

The  interest  term  is  200  da.,  or  6  mo.  20  da. 

ORAL   EXERCISE. 

Find  the  time,  using  1  day.  Find  the  time  using  1  month. 


Prm. 

Int. 

Rate. 

Prin. 

Int. 

Rate, 

1. 

$  600 

$1.50 

6%. 

5.     $  420 

$  6.30 

6%. 

2. 

$1200 

$6.00 

9%. 

6.     $  800 

$90. 

6%. 

3. 

$720 

$1.32 

6%. 

7.     $1200 

$80. 

5%. 

4. 

$3000 

$7.50 

6%. 

8.     $2000 

$75. 

9%. 

9. 

In   what 

time  does 

a  princii 

pal  double  itself 

(simple  int€ 

Test)  at 

4%,  6%,  6%,  8%,  7%,  10%,  9%,  3%? 


EXERCISE. 


1.  On  January  25,  $7200  is  invested  at  6%  simple  interest  to  remain 
until  it  amounts  to  $7500.     Find  the  interest  period. 

2.  On  what  date  will  $3200,  invested  July  8,  1909,  at  6%,  amount  to 
$5000? 

Periodic  Interest. 

Periodic  interest  is  the  interest  on  a  principal,  payable  at 
stated  intervals,  plus  simple  interest  on  overdue  interest 
payments.  It  can  be  legally  enforced  in  but  few  states,  but 
it  is  collected  indirectly  by  means  of  non-interest  notes,  given 
for  the  regular  payments,  which  draw  simple  interest  when 
overdue. 

Illustration.  What  is  due  after  one  year  and  six  months  on  a  $1200 
loan,  at  6%  interest,  payable  quarterly,  nothing  having  been'  paid? 


INTEREST.  301 

Solution. 

Interest  on  $1200  for  3  mo.  =  $18,  the  quarterly  payment. 
Interest  on  $1200  for  1  yr.  6  mo.  =  $108,  the  total  sunple  payments. 
At  the  end  of  1  yr.  6  mo., 

The  first      payment  is  overdue  1  yr.  3  mo. 
"    second        "         "        "        1    "   0    " 
"    third        ,"        "        "  9    " 

"    fourth      '"         "        "  6    " 

u      g£^i^  a  u  u  3     " 

The  sum  of  overdue  payments  3  yr.  9  mo. 

The  total  overdue  interest  on  the  several  equal  payments  equals  the 
interest  on  one  payment  for  the  total  time,  or  for  3  yr.  9  mo. 
Interest  on  $18  for  3  yr.  9  mo.  =  $     4.05 
Regular  interest  payments,  108. — 

Principal,  1200. 

Amount  due,  $1312.05 

EXERCISE. 

1.  On  June  1,  1908,  $3000  was  lent  at  5%,  payable  semi-annually. 
Nothing  having  been  paid,  what  sum  is  due  in  full  settlement,  on  December 
1,  1909? 

2.  Find  the  amount  due  in  full  settlement  for  a  loan  of  $7000,  at  6%, 
payable  quarterly,  on  which  nothing  has  been  paid  for  a  term  of  four  years? 

3.  Repeat  example  2,  on  a  basis  of  semi-annual  payments. 


CHAPTER  XXXVI. 

COMPOUND  INTEREST. 

Compound  interest  is  interest  computed  on  the  sum  of  a 
principal  and  its  due  and  unpaid  interest.  Interest  must  be 
"  compounded,"  or  added  to  the  principal,  at  stated  intervals, 
as  quarterly,  semi-annually,  or  annually.  Other  periods  may 
be  fixed  by  agreement.  While  not  recoverable  by  law,  in 
most  states,  the  accepting  of  compound  interest  does  not 
constitute  usury.  If  a  new  obligation  is  given  at  the  end  of 
any  of  the  fixed  periods  for  the  amount  then  due,  the  obli- 
gation is  binding. 

Compound  interest  is  used  by  savings  banks,  in  reckoning 
interest  on  deposits,  by  insurance  companies  and  by  business 
men  generally,  in  computing  the  cumulative  value  of  invest- 
ments at  a  fixed  rate,  where  earnings  are  not  withdrawn. 

Illustration.    $12,000  invested  at  6%,  and  compounded  semi-annually 
for  two  years,  will  amount  to  what  sum? 
Solution. 

$12,000       =  original  principal. 

360       =  interest  for  6  mo.  (3%). 
12,360       =  amount,  end  of  6  mo. 

370.80  =  interest  for  6  mo.  on  last  amount  (3%). 


12,730.80  =  amount,  end  1  yr. 

381.92  =  interest  for  6  mo.  on  last  amount  (3%). 


13,112.72  =  amount,  end  1  yr.  6  mo. 

393.38  =  interest  for  6  mo.,  on  last  amount  (3%). 


$13,506.10  =  final  amount. 

ORAL    EXERCISE. 

1.  The  interest  for  4  years  at  8%,  compounded  semi-annually,  equals 
the  interest  for  ?  semi-annual  periods,  at  ?  %. 

2.  In  compounding  quarterly  for  8  years  at  6%,  there  are  ?  interest 
periods  at  ?  %  per  period. 

302 


COMPOUND   INTEREST. 


303 


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304  BUSINESS  ARITHMETIC. 

3.  Compute  the  amount  of  a  $4000  savings  deposit,  compounded 
quarterly  for  one  year,  at  4%  per  annum. 

EXERCISE. 

1.  A  fund  of  $16,000  at  6%,  compounded  semi-annually,  amounts  in 
5  years  to  $ — .    , 

2.  $5000  invested  Jan.  1,  1906,  at  4%,  compounded  quarterly,  will 
amount  to  $ on  March  21,  1907. 

Note.     Compute  interest  for  fractional  term  on  last  full  amount. 

Compound  interest  is  computed,  as  a  rule,  from  tables  similar 
to  the  one  on  page  303. 

Illustration.     Find  the  amount  of  $2500  for  6  years  at  6%,  com- 
pounded quarterly. 
Solution. 

In  6  years  there  are  4  X  6,  or  24,  interest  periods. 
The  rate  of  6%  per  year  equals  1^%  per  quarterly  period. 
The  problem  is  to  find  the  amount  of  $2500  for  24  periods  at  1^%. 
$1.429503  =  amount  of  $1.00  for  24  periods  at  1^%. 
2500 
$3573.76         =  amount  of  $2500  for  24  periods  or  6  years. 

It  is  evident  that  time  and  rate  should  be  reduced  to  periods 
and  period  rates,  and  that  the  amount  of  $1.00  for  this  number 
of  periods  should  be  multiplied  by  the  number  of  dollars  in 
the  principal.  If  the  number  of  periods  is  greater  than 
twenty-five,  find  the  amount  for  twenty-five  periods,  and  then 
find  the  amount  of  this  amount  for  the  remaining  periods. 

ORAL   EXERCISE. 

Read  from  the  table  the  amount  of  $1000  if  compounded: 

1.  Quarterly  at  6%  per  annum  for  6  years. 

2.  Semi-annually,  at  8%  for  5  years. 

3.  Bi-monthly,  at  6%  for  3  years. 

4.  Every  4  months,  at  9%  for  6  years. 

EXERCISE. 

Compute  the  amount  of 

1.    $5950,  for  3  yr.  6  mo,  at  5%,  compounded  semi-annually. 


COMPOUND  INTEREST.  305 

2.  $12,600  for  6  yr.  6  mo.,  at  8%,  compounded  quarterly. 

3.  $7200,  from  July  1,  1904,  to  October  27,  1909,  at  6%,  compounded 
semi-annually. 

Compute  the  compound  interest  on 

4.  $1200  for  8  yr.  4  mo.  at  6%,  compounded  quarterly. 

5.  $2500  for  2  yr.  6  mo.,  at  9%,  compounded  every  4  months. 

6.  $3250  from  October  1,  1909,  to  November  10,  1913,  at  5%  com- 
pounded 33mi-annually. 

What  sum  must  be  invested  at 

7.  6%,  compounded  quarterly,  to  amount  to  $7500  in  6  years? 

8.  8%,  compounded  semi-annually,  to  amount  to  $12,500  in  4  yr.  6  mo.? 

9.  9%,  compounded  semi-annually,  to  amount  to  $8000  in  3  yr.  6  mo.? 


21 


CHAPTER  XXXVII. 

LOANS  AND  PAYMENTS. 

INTRODUCTORY    EXERCISE. 

1.  What  may  lead  one,  in  his  private  capacity,  to  borrow  money? 

2.  Name  objections  to  such  borrowing. 

3.  What  may  cause  a  business  man,  or  organization,  to  borrow? 

4.  Does  a  prosperous  business  ever  need  a  loan? 

5.  When  might  a  business  borrow  in  order  to  pay  cash  for  goods  bought 
on  credit? 

6.  How  may  a  loan  spread  the  cost  of  an  improvement,  for  which  it 
pays,  over  a  long  period? 

7.  What  is  the  danger  of  lending  money  on  a  verbal  promise  to  repay? 

8.  What  business  paper  commonly  is  used  as  evidence  of  a  loan,  or  of 
a  payment  due  in  the  future? 

9.  How  else  may  loans  be  "secured"? 

ORAL    EXERCISE. 

1.    I  borrow,  to-day,  $1200  for  3  mo.  at  6%.     I  receive  $  ?,  but  must 

pay  back  $  ?,  paying  %  ?  for  the  use  of  the  money. 
Determine  interest  and  amount  due  on  these  loans: 


Loan. 

Period. 

Rate. 

Loan. 

Period. 

Rate. 

2. 

$  800 

9  mo. 

6%. 

6.     $2500 

24  da. 

6%. 

3. 

1200 

36  da. 

5%. 

7.        540 

30  da. 

6%. 

4. 

400 

5  mo. 

9%. 

8.         250 

2yr. 

4%. 

5. 

1500 

2  mo. 

4%. 

EXERCISE. 

1.  What  interest  has  accrued  to  date,  at  4^%,  on  a  loan  contracted 
Feb.  11  of  last  year,  for  $820? 

2.  On  Mar.  18,  I  must  meet  a  loan  of  $840,  contracted  Oct.  21  last, 
and  drawing  6%  interest.  If  I  have  $520,  what  additional  sum  must 
I  raise? 

306 


LOANS  AND  PAYMENTS.  307 

3.  Settlement  of  a  3%  loan  of  $90,000,  dated  Jan.  20,  is  demanded 
Jan.  25.     What  is  due? 

4.  A  3%  loan  of  $6000,  payable  on  demand,  is  contracted  on  Feb* 
ruary  16,  and  renewed  on  February  25  at  4%.  Settlement  is  made,  March 
2,  by  paying  $— . 

5.  My  brokers  lend  for  me,  on  a  commission  of  1/2%  interest: 

Jan.  19,  to  C.  Warren,  $900  at  6%. 
Feb.  3,  to  M.  C.  Poston,  $1250  at  5%. 
Feb.  16,  to  R.  Fallon,  $1920  at  6%. 
Determine  my  net  interest  on  the  following  January  1. 

PROMISSORY  NOTES. 
A  promissory  note  is  a  written  promise  to  pay  a  specified 
sum  of  money,  on  demand,  at  a  specified  time,  with  or  without 
interest.     It  is  a  common  evidence  of  debt. 

Illustration. 

%560xx  Cincinnati,  0.,  Mar.  21,  19/5. 

Sixty  days after  date  I  promise  to  pay 

to  the  order  of James  Fielding 

at 7S6  Fifth  Ave 

Five  hundred  sixty  X2;/100 Dollars 

Value  received  with  Interest  at  ^  %  per  annum. 

Robert  C.  Evans 
No.  97.... Due. May  W.... 

Note.  Robert  C.  Evans,  who  signs  this  note,  is  the  maker;  James 
Fielding  is  the  payee.  If  the  note  had  read  "On  demand,  I  promise," 
etc.,  it  would  have  been  payable  whenever  presented  to  Evans. 

Notes  are  negotiable,  if  transferable  by  the  payee  to  other 
parties  by  endorsement  of  his  signature  on  the  back.     To  be 

transferable,  notes  must  read   "  Pay  to  ,  or   order," 

"  Pay  to  the  order  of ,"  or  "  Pay  to ,  or  bearer." 

If  the  words  "  or  order,"  or  "  or  bearer"  are  omitted  the  note 
is  not  transferable. 

The  following  are  the  common  forms  of  endorsement: 
(1)  Endorsement  in  Blank.         (1)  James  Fielding. 
This  transfers  right  of  owner- 
ship to  the  bearer. 


308  BUSINESS  ARITHMETIC. 

(2)  Full  Endorsement.  (2)  Pay  to  the  order  oi 
Here  James  Fielding  transfers     Robert  Anderson, 

the  note  to  Robert  Anderson,  James  Fielding, 

but  since  the  expression  "pay 
to  the  order  of"  is  used, 
Anderson  may  transfer  the 
note  to  others. 

(3)  Qualified  Endorsement.  (3)  Pay  to  the  order  of  C. 
The  note  is  transferred  by  P.  Rankin,  without  recourse 
Fielding  to  Rankin,  but  the     to  me 

former  by  use  of  the  phrase  James  Fielding. 

"without  recourse"  refuses  to 

agree  to  pay  the  note  when 

due,  in  case  the  maker  fails  to 

pay. 

The  last  legal  holder  of  the  note  secures  payment  from  the 
maker  at  maturity.  If  the  maker  fails  to  pay,  each  endorser 
in  turn,  unless  his  endorsement  is  qualified,  may  be  held  for 
payment.  In  case  of  non-payment,  the  note  may  be  given  to 
a  notary  public,  who  formally  demands  payment.  If  payment 
is  refused,  he  legally  protests  the  note,  issuing  notice  to  each 
indorser  and  thus  binding  them  to  payment.  This  protest 
is  a  basis  for  legal  action  to  compel  payment.  The  amounts 
collectible  are  the  face,  interest  if  any,  and  legal  charges. 
Non-interest-bearing  notes  draw  interest  after  due. 

Notes  signed  by  two  or  more  parties  and  reading  "  we  jointly 
promise,"  are  termed  joint  notes,  each  party  being  held  for 
his  proportional  part  of  the  note  (see  p.  314).  If  the  note 
reads  "  We  jointly  and  severally  promise,"  each  signer  is  held 
for  full  settlement.  Some  states  hold  joint  notes  as  similar 
to  Joint  and  several  notes. 

Collateral  notes  contain  a  permit  to  enable  the  holder,  in 
case  of  failure  to  pay  at  maturity,  to  dispose  of  certain  property 
(collateral)  of  the  maker,  and  placed  in  possession  of  the  payee, 


LOANS  AND  PAYMENTS.  309 

as  security,  at  the  time  of  giving  the  note.  One  form  of 
collateral  note  is  shown  on  page  346. 

When  a  loan  is  made  on  real  estate,  notes  usually  are  given, 
secured  by  a  mortgage  or  deed  of  trust,  under  which  the 
property  may  be  disposed  of  in  order  to  meet  payments. 
Receipts  for  goods  in  warehouse  or  grain  elevator  likewise, 
may  be  endorsed  to  the  lender  as  security  for  loans. 

The  bonds  of  corporations  are  simply  corporate  notes,  and 
frequently  are  secured  by  mortgages. 

A  commercial  draft  is  an  order  written  by  one  person  re- 
questing a  second  person  to  pay  a  stated  sum  of  money  to  the 
order  of  the  former,  or  to  a  third  person.  The  parties  to  a 
draft  are  the  drawer,  or  maker,  the  drawee,  or  person  drawn  on, 
and  the  payee.  When  accepted  by  the  drawee,  commercial 
drafts  usually  are  negotiable.  Accepted  drafts  are  termed 
acceptances. 

Illustration  (1).  Andrews  sells  Parker  $646  worth  of  merchandise, 
on  terms  of  "60  day  acceptance."  He  sends  Parker  an  mvoice,  accom- 
panied by  the  draft  shown  on  page  310.  Parker,  finding  the  goods  satis- 
factory, writes  "Accepted"  across  the  face  of  the  draft  and  signs  his  name, 
thus  binding  himself  to  pay  the  paper  when  due.  He  then  returns  the  draft 
to  Andrews.  By  writing  "Accepted,  Payable  at  the  First  Nat.  Bank, 
Chas.  Parker,"  Parker  could  have  turned  the  settlement  over  to  his  bank. 
Practically,  the  draft  is  now  a  note. 

Illustration  (2).  Andrews  might  wish  to  transfer  payment  to 
Henry  Connors,  a  creditor  of  his.  In  that  case  he  might  make  out  the 
above  draft  and  transfer  it  by  endorsement,  or  he  might  make  out  the 
draft  in  the  form  shown  on  page  311.  In  the  latter  case  he  would  prob- 
ably send  the  draft  to  Connors  and  the  latter  would  present  it  to  Parker  for 
acceptance. 

When  drafts  read  "after  sight,"  in  place  of  "after  date,"  the  acceptance 
must  be  dated,  and  maturity  counts  from  that  date.  Drafts  reading 
simply  "at  sight,"  are  payable  at  once  without  acceptance.  They  are 
often  drawn  by  merchants  against  customers,  and  are  sent  through  a  bank 
for  collection.  If  the  terms  of  a  sale  read,  "subject  to  draft  after  30  days" 
it  is  understood  that  a  sight  draft  will  be  drawn  at  the  end  of  the  period. 


310 


BUSINESS  ARITHMETIC. 


LOANS  AND  PAYMENTS. 


311 


312  BUSINESS  ARITHMETIC. 

Factors  Affecting  the  Maturity  of  Negotiable  Paper. 

The  wording  of  negotiable  paper  affects  the  date  of  maturity. 

On  papers  reading  "  —  months  after  date  "  and  " days  after 

date"  maturity  is  reckoned  from  the  date  of  the  note,  the 
former  by  approximate  and  the  latter  by  exact  time.  A  few 
states  still  allow  "  three  days  of  grace,"  after  date  of  maturity, 
before  payment  must  be  made. 

If  drafts  read  "  after  sight,"  or  "  at  —  days  sight,"  the 
maturity  is  reckoned  from  the  date  of  acceptance.  Thus  the 
draft  on  page  311  is  payable  60  days  from  March  20. 

After  the  calendar  date  of  maturity  has  been  reckoned,  the 
actual  payment  date  must  be  determined  with  due  regard  to 
the  varying  state  laws  concerning  Sundays,  holidays,  etc. 
Thus,  in  Pennsylvania,  a  note  coming  due  Saturday  is  not 
payable  until  the  following  Monday,  while  if  Monday 
should  happen  to  be  a  holiday,  it  would  become  payable  on 
Tuesday. 

Summary  of  Laws  Relating  to  the  Maturity  op  Negotiable 
Instruments. 

Note.  These  laws  are  in  process  of  change,  in  many  states.  Ask  a 
banker  for  the  latest  regulation. 

For  states  allowing  days  of  grace,  or  for  local  laws,  consult  a  local 
banker  or  business  man. 

Requirement.  State  or  Country. 

Due  Sunday  or  holiday,  pay    Ala.,  Ariz.,  Cal.,  Col.,  D.C.,  Fla.,  Hawaii, 

next  business  day.  111.,  Ind.,  la.,  Kas.,  Ky.,  La.,  Me.,  Minn., 

•Mo.,    Mont.,    N.H.,    N.M.,    Okla.,    S.C, 

S.Dak.,  Tex.,  W.Va.,  Wis.,  Canada. 

Due  Sunday  or  holiday,  pay    Alaska,  Ark.,  Del.  (Kent  and  Sussex  Co.), 

preceding  business  day.  Ga.,  Miss.,  Nev.,  Philippines,  P.R.,  Cuba, 

Mexico. 

Due   Saturday,    Sunday    or    Conn.,   Idaho,   Md.,    Mass.,    Neb.,    N.J., 

holiday,  pay  next  business    N.Y.,  N.C.,  N.Dak.,  O.,  Oreg.,  Pa.,  R.I., 

day.  Mich.,  Tenn.,  Ut.,  Vt.,  Va.,  Wash.,  Wyom. 


LOANS  AND  PAYMENTS.  313 

Holidays. 

Day^ Country  or  State. 

January  1.  All  states    (except   Mass.  and  Kansas), 

Canada,  Cuba,  and  U.  S.  possessions. 
February  22.  All  states  and  U.  S.  possessions. 

May  30.  All  states  and  possessions,  except  Ala., 

Ark.,  Fla.,  Ga.,  La.,  Miss.,  N.  C,  Okla.,  S. 
C.  and  Tex. 
July  4.  All  states  and  U.  S.  possessions. 

Labor   Day    (first    Monday        All  states    (except    Md.,  Wis.,  Wyom.) 

in  September).  and   U.  S.  possessions,  except  Philippines. 

Thanksgiving  Da}'^  (by  proc-        All  states  and  possessions. 

lamation,       usually      last 

Thursday  in  November). 
December  25  (Christmas) .             All  states  and  U.  S.  possessions,  Canada 
and  Cuba. , 

Many  states  or  counties  have  local  holidays  other  than  those  named  above. 

EXERCISE. 
Find  the  dates  of  maturity  of  this  commercial  paper,  consulting  the 
above  summary.     Use  a  calendar  and  allow  for  hoUdays,  etc. 

Maturity  in  Your  Own  State 

1.  90  day  note,  dated  January  27. 

2.  60  day  note,  dated  Dec.  24. 

3.  "Three  months  after  date"  paper,  dated  Mar.  4. 

4.  "Sixty  days  after  date"  draft,  dated  June  14,  accepted  June  17. 

5.  "At  90  days'  sight"  draft,  dated  Aug.  31,  accepted  Sept.  3. 

Maturity  in  Different  States. 

6.  "60  days'  sight"  draft,  dated  May  4,  accepted  May  5,  payable  in 
Minnesota. 

7.  A  "90  days  after  date"  draft   dated  September  3,  accepted  Sep- 
tember 4,  payable  in  Ohio. 

8.  A  4  months  note  date  Mar.  4,  1909,  payable  in  Texas. 

9.  A  120  day  note,  dated  April  24,  1909,  payable  in  Canada. 

QUESTIONS    FOR    DISCUSSION. 

1.     As  a  rule,  which  allow  a  longer  time  for  payment,  notes  given  for 
months,  or  for  the  same  number  of  thirty  day  periods? 


314  BUSINESS  ARITHMETIC. 

2.  Is  thirty  day  paper  ever  better  for  the  payer  than  one  month  paper? 

3.  Give  reasons  for  allowing  days  of  grace.  Why  are  they  being 
abolished? 

4.  Ordinarily,  do  the  regulations  as  to  Sundays  and  holidays  favor 
payer  or  payee? 

5.  Why  should  the  date  of  payment  be  fixed  beyond  question 

EXERCISE. 

1.  What  yearly  saving  in  interest  results  from  refimding  at  3|%, 
$52,500  worth  of  5%  bonds? 

Note.     Refunding  means  re-issuing  instead  of  paying  at  maturity. 

2.  What  sum  should  be  paid  on  June  20,  in  full  payment  for  a  90  day 
non-interest  note  for  $1200,  dated  February  11?     Interest,  6%. 

3.  What  sum  will  settle  this  note  to-day : 

$520xY?r  Washington,  D.  C,  August  12,  1909. 

On  demand,  after  date,  we  jointly  promise  to  pay  to 

the  order  of  C.  M.  Fairley 

Five  hundred,  twenty  ^ Dollars. 

Value  received,  with  interest  at  6%  per  annum. 

A.  B.  Morton, 
Henry  C.  Mason. 

4.  I  lend  C.  P.  Rogers  $2260,  at  5%,  taking  a  first  mortgage  on  his 
farm.     What  quarterly  payment  should  he  make? 

At  the  end  of  two  years,  nothing  having  been  paid,  the  mortgage  is 
foreclosed,  and  the  property  sold  for  $3675,  the  charges  for  the  sale  being 
$61.90.     What  is  due  Rogers? 

Income  and  Outgo. 

5.  An  investment  of  $19,700  in  5%  mortgages  yields  a  semi-annual 
income  of  $ — . 

6.  A  broker  lends  for  a  customer  $18,000  at  5%,  charging  a  com- 
mission of  5%  of  the  first  year's  income.  What  is  the  client's  net  income 
the  first  year? 

7.  On  February  19,  I  borrowed  $12,000,  at  3i%,  giving  my  demand 
note.  On  March  3, 1  lent  the  money  at  5%.  On  December  15,  my  note 
being  presented  for  payment,  I  called  in  my  loan.     Net  gain? 

Partial  Payments. 
Partial  payments  may  be  made,  by  agreement,  before  or 
after  the  maturity  of  a  note  or  bond.     Such  payments  should 


LOANS  AND  PAYMENTS. 


315 


be  endorsed  on  the  back  of  the  paper  in  one  of  the  forms  shown. 


Face. 
$1800.1^^    Washington,  D.  C,  June  10,  1909. 

One  year  after  date  I  promise  to  pay 

to  the  order  of  C.  P.  Crandall 

Eighteen  hundred Dollars. 


With  interest  at  6%  per  annum. 

Robert  Davis. 


Back. 
Paid,  July   12,  1909,  $200. 
Paid,  Oct.    12,  1909,  $300. 
Paid,  Mar.  10,  1910,  $400. 


Other  forms  of 

Indorsement. 

(1) 

July  12,  1909,  paid  $200. 

(2) 
Rec'd,  7/12/09,  $200. 
(3)      ^ 
Rec'd    on    the    within 
note,  July  12,  1909,  three 
hundred  dollars. 

Question.  Why  is  no  signature  to  an  endorsement  of  a  partial  pay- 
ment necessary? 

The  partial  payment  introduces  several  factors  into  the 
computation  of  the  amount  due,  the  question  arising  as  to 
whether  payments  shall  be  applied  to  principal  or  to  interest, 
or  to  both.  State  laws  govern  the  matter,  but  many  states 
have  now  legalized  the  United  States  rule,  established  by  the 
Supreme  Court  of  the  United  States.  The  so-called 
Merchants'  rule  is  also  frequently  used.  The  United  States 
rule  is  very  commonly  used  for  long  term  paper. 


INTRODUCTORY    EXERCISE. 

1.  What  is  due  at  the  end  of  6  months  on  a  6%  note  for  $1000? 

2.  If  $230  is  then  paid,  what  sum  will  pay  the  interest  due?  What  is 
left  to  reduce  the  face?  How  much  of  the  debt  is  still  outstanding?  On 
what  sum  should  interest  now  be  paid? 

3.  What  interest  is  due  6  months  later?     What  amount? 

4.  If,  now,  $324  is  paid,  the  new  principal  becomes  $ . 

5.  The  debt  is  finally  settled  a  year  later,  by  paying  $ . 

Illustration.    United  States  Rule. 
Example.     Find  the  balance  due  at  maturity,  on  the  note  shown  above. 


316  BUSINESS  ARITHMETIC. 

Solution. 

Face  of  note $1800. — 

Interest,  June  10  to  July  12,  2  mo.  2  da 18.60 

Amount  due,  July  12 1818.60 

Payment,  July  12 200.— 

New  Principal 1618.60 

Interest  on  new  principal,  July  12  to  Oct.  12,  3  mo 24.28 

Amount  due,  Oct.  12 1642.88 

Payment,  October  12 300.— 

New  principal,  balance  due 1342.88 

Interest  on  new  principal,  Oct.  12  to  Mar  10,  1910,  4  mo.  28  da. . .       33.12 

Amount  due.  Mar.  10 1376. — 

Payment,  Mar.  10 400.— 

New  principal,  balance  due 976. — 

Interest  from  Mar.  10  to  maturity,  Jime  10,  3  mo 14.64 

Amount  due  at  maturity $  990.64 

Note.  Under  this  rule,  a  payment  is  first  applied  to  pay  off  interest, 
and  then  to  reduce  principal.  In  case  any  payment  is  less  than  the  interest 
accrued,  the  principal  remains  unaltered  untU  such  a  date  that  the  sum  of 

Eayments  exceeds  the  interest  accrued.     For  example,  if  the  first  payment 
ad  been  $10,  it  would  have  been  added  to  the  following  $300  payment, 
and  interest  found  on  the  original  face  from  June  10  to  October  12. 

Question.  What  would  be  the  result  of  using  a  payment  jess  than  the 
accrued  interest? 

EXERCISE. 
Use  approximate  time  for  ex.  1-3;  exact  time  for  ex.  4-5. 

1.  Date  of  loan,  June  26,  1908;  amount,  $3000;  interest  6%.  Pay- 
ments: Aug.  15,  1908,  $450;  June  21,  1909,  $800;  Aug.  21,  $300;  July  16, 
1909,  $425.     Fmd  the  balance  due  Oct.  21,  1909. 

2.  A  two  year  note  for  $3760,  dated  Oct.  15,  Paid,  11/8/11,  $400 
1911,  and  bearing  6%  interest,  has  these  endorsed  "  12/9/11,  $200 
payments.     What  is  due  at  maturity?  "       7/3/12,  $600 

"      11/6/12,  $  20 
"       1/7/13,  $520 

3.  On  Oct.  27,  Robert  Evans  paid  $1200  on  a  demand  note  for  $3000, 
dated  Jan.  17,  and  bearing  6%  interest.  On  Nov.  5,  he  paid  $200,  and 
gave  a  new  3  mo.  note  for  the  balance  due.  Determine  the  face  of  the 
new  note. 

4.  Determine  the  amount  due  on  this  note  at  maturity. 

$3500.—  Baltimore,  Md.,  July  15,  1911. 

Fifteen  months  after  date,  I   promise  to  pay  to  the  order  of 

Elton  Howells,  thirty-five  hundred Dollars. 

Value  received,  with  interest  at  5%.  Jambs  Elwbll. 


LOANS  AND  PAYMENTS  317 

Payments  endorsed  as  follows: 

Paid,  Nov.    5,  '11,  $  800. 
"     Mar.  17,  '12,  $  250. 
"     Nov.  12,  '12,  $1500. 
5.    From  this  Loan  Book  record,  find  the  balance  due,  Oct.  1,  1912, 
at  3%: 

Loan  to  C.  P.  Davidson, 
NoJ  Date.  Amount.        Date  Payment.        Amount  Payment- 

136         Oct.  1,  '10        $40,000  Jan.    15,  '11  $5,000. 

Mar.  20,  '11  10,000. 

Aug.  21,  '11  10,500. 

Jan.     1,  '12  3,000. 

INTRODUCTORY    EXERCISE. 

1.  What  is  due  at  maturity  on  a  6  months  note  for  $1200,  dated  June  1, 
and  bearing  6%  interest? 

2.  On  Oct.  1,  a  payment  of  $600  was  made.  If  this  payment  was 
allowed  to  draw  interest,  to  what  would  it  amount  when  the  note  was  due? 

3.  What  is  the  difference,  at  maturity,  between  the  value  of  the  note 
and  the  value  of  the  payment? 

The  Merchant's  rule  is  commonly  used  in  banks  and  com- 
mercial houses,  especially  for  short  term  notes. 

Illustration.     Merchant's  Rule. 

Example.     Determine  the  balance  due  on  the  note,  page  315. 

Solution. 

Face  of  note $1800. — 

Interest  to  maturity — 1  year 108. — 

Amount  due  at  maturity $1908. — 

First  pajonent $200. — 

Interest  on  payment  to  maturity,  July  12,  1909,  to 

June  10,  1910,  10  mo.  28  da 10.93 

Second  payment 300. — 

Interest,  October  12  to  June  10, 1910,  7  mo.  28  da.     1 1 .90 

Third  payment 400.— 

Interest,  Mar.  10  to  Jime  10,  3  mo 6. — 

Total  value  of  payments  at  maturity $928.83  928.83 

Difference  between  value  of  note  and  of  payments  $  979.17 

Note.  Under  this  method,  the  face  draws  interest  until  the  date  of 
settlement,  and  each  payment  is  allowed  interest  until  the  date  of  settle- 
ment or  maturity.  The  balance  due  is  the  difference  at  time  of  settlement, 
between  the  value  of  the  face  and  the  value  of  the  payments. 


318 


BUSINESS  ARITHMETIC. 


EXERCISE, 
Compute  by  both  rules,  the  balance  due  March  29,  1912,  on  this 


1. 

note. 

$2400.—     Cincinnati,  Ohio,  Sept.    7,  1911. 

On  demand,  after  date,  I  promise  to  pay- 
to  the  order  of  S.  P.  Campbell 

Twenty-four  hundred Dollars 

with  interest  at  6%  per  annum  until  paid. 
A.  B.  Norton. 

Solve  the  following,  using  exact  time. 


Rec'd,  Nov. 
Rec'd,  Jan. 
Rec'd,  Mar. 


5/11,  $300. 
16/12,  $580. 
3/12,  $627.50 


Loan  No. 

Date 

Name 

Amount 

Rate 

Date  Pay. 

Am't 

Settled 

2 

1911 
Oct.    8 

James  Baker 

$36,000 

3% 
6% 
6% 

Nov.    5/11 
Dec.  21/11 
Mar.  20/12 

$4000 
6000 
8000 

June 
21/12 

3 

Oct.  11 

Robert  Jones 

$4,500 

Jan.    15/12 
Mar.  12/12 
Apr.  21/12 

$  400 
1150 
1200 

July 
17,  '12 

4 

Oct.  15 

Henry  Sanders 

$1850 

Dec.  20/11 
Jan.    15/12 
Feb.  20/12 
Apr.  14/12 

$200 
185 
300 
400 

May 
21/12 

5-9.    Solve  all  the  examples  given  under  the  U.  S.  rule  by  the  Mer- 
chants' rule. 

QUESTIONS    FOR    DISCUSSION. 

1.  Compare  the  illustrative  examples  for  the  two  rules  and  describe 
the  different  treatment  of  payments. 

2.  How  are  interest  periods  obtained  under  the  U.  S.  rule?    Under 
the  Merchants'  rule? 

3.  Why  does  it  make  no  difference,  under  the  Merchants'  rule,  whether 
the  payments  are  greater  or  less  than  the  interest? 

4.  Explain  the  reason  for  the  difference  in  the  final  amount  due  under 
the  two  rules. 

Bank  Discount. 
A  dealer  buys  "on  credit"  in  order  that  he  may  dispose  of 
his  goods,  in  part  at  least,  before  the  bill  for  them  becomes  due. 
He  thus  makes  them  "pay  for  themselves"  and  lessens  the 
amount  of  his  necessary  capital.  The  manufacturer,  however, 
pays  cash  for  his  heavy  labor  expense,  and  for  some  supplies. 


LOANS  AND  PAYMENTS.  319 

Were  he  obliged  to  wait  for  payments  from  customers,  while 
paying  out  cash,  he  might  be  forced  to  carry  extra  heavy  capi- 
tal, on  which  he  could  not  earn  a  fair  return  on  investment. 
To  avoid  this,  he  takes  notes  from  his  customers,  or  drafts 
on  them,  and  sells  these  to  a  commercial  bank,  at  a  slight 
discount.  This  discount  is  computed  as  interest  for  the  time 
the  note  has  to  run.  The  bank  collects  the  full  amount  at 
maturity,  from  the  maker,  the  discount  representing  its  profit, 
while  the  manufacturer,  although  selling  on  credit,  gets  the 
immediate  use  of  most  of  his  money. 

Commercial  banks  are  chartered  by  law  to  receive  and  lend 
money  and  to  make  collections.  A  large  part  of  the  income 
of  many  banks  is  obtained  by  loans  through  discounts.  The 
bank's  directors,  or  some  designated  officials,  examine  all 
paper  offered  for  discount,  as  to  its  makers,  endorsers,  or 
security.  Some  banks  discount  only  for  their  regular  custo- 
mers. In  such  cases,  the  loan,  or  proceeds,  is  placed  to  the 
credit  of  the  discounter,  who  "checks  out"  the  money  as 
needed.  The  bank  gains  slightly  from  the  extra  use  of  the 
deposited  money. 

QUESTIONS    FOR    DISCUSSION. 

1.  Of  what  advantage  is  it  to  a  dealer  to  buy  on  credit,  rather  than  for 
cash? 

2.  What  are  the  objections  to  selling  "on  credit"? 

3.  What  is  the  disadvantage  of  buying  goods  for  cash  and  selling  on 
time? 

4.  What  is  meant  by  "tying  up"  one's  capital? 

Discount  Terms  and  Rules. 

The  term  of  discount  is  the  time  between  date  of  maturity 
and  date  of  discount.  A  note  or  draft  is  discounted  for  its 
"full  term"  when  it  is  discounted  for  the  time  stated  in  it. 

The  amount  or  value  at  maturity  is  the  face,  plus  interest, 
if  any. 


320  BUSINESS  ARITHMETIC. 

The  discount  is  the  sum  deducted  by  the  bank. 

The  proceeds  is  the  sum  credited,  or  paid,  to  the  person 
discounting. 

Commercial  banks  vary  greatly  in  their  rules  for  discounting. 
Some  use  exact  time,  others  approximate  time;  some  have  a 
minimum  discount  of  25c,  etc.  The  general  rules  given  below 
may  be  modified  to  conform  to  local  custom: 

1.  Discount  is  reckoned  on  the  amount  due  at  maturity. 

2.  Notes  discounted  for  full  terms  are  discounted  for  the 
time  stated  in  them. 

3.  Notes  discounted  for  partial  terms  are  discounted  for  the 
exact  number  of  days  between  the  date  of  discount  and  date 
of  maturity. 

Exception.  When  interest-bearing  notes  are  discounted,  at  the  same 
rate  as  the  interest  rate,  for  a  full  term,  the  proceeds  equal  the  face. 

Illustrations.  (1)  NonAnterest  note.  Discounted  for  full  term  at  6%. 
$840^ji  Philadelphia,  Pa.,  Jan.  27,  190— 

Three  months after  date .  .  / .  .  promise  to  pay  to  the 

order  of James  C.  Leonard 

at The  Cross  National  Bank 

Eight  hundred  forty Dollars 

Value  received  with  Interest  at  — %  per  annum. 

Chas.  R.  CarleUm. 
^o.  89 . . .  .Bue  Apr.  27 
Solution. 

Discount  term,  3  mo.  $840.00= face. 

Amount  due  at  maturity,        $840. —       $     8.40=  discount,  2  mo. 
Discoimt,  for  3  mo.  12.60  4.20= discount,  1  mo. 

Proceeds,  $827.40      iri2.60= discount,  3  mo. 

(2)  The  same  note,  discounted  February  9. 
Date  of  maturity,  April  27. 
Discount  term,  February  9  to  April  27,  77  days. 

Amount  due,  $840. —  $840. — =base,  for  discount. 

Discount,  10.78  $    8.40  =  discount  for  60  da. 

Proceeds, 


$829.22 

$     1.40  = 

(( 

"  10   " 

.84  = 

« 

a      g    « 

.14  = 
$  10.78  = 

((      -t     ft 

"Y7  " 

LOANS  AND  PAYMENTS. 


321 


(3)  Interest  note.    Discounted  for  full  term  at  6%. 

$784^f^  Cleveland,  0. 

Sixty  days after  date ..../. 

to  the  order  of  ....  Cha^.  Morgan  &  Co. 

at. . .  .1685  Payne  Ave. 

Seven  hundred  eighty-Jour 


October  U,  190— 
. .  .promise  to  pay 


.Dollars 


Value  received  with  Interest  at  6%  per  annum. 

Henry  T.  Carmody 
No.  9   Due  Dec.  IS  1685  Payne  Ave. 

Solution.     See  rule  3,  exception.     The  proceeds  equal  $784. — 
(4)  The  same  note,  discounted  for  full  term  at  8%. 
Solution. 

Face  of  note  $784. — 

Interest,  60  da.,  6%  7.84 


Discount  term,  60  da. 
$791.84     =base  for  discount. 


Amount  at  maturity,    $791.84 

$     7.9184  = 

discount. 

60  da.  6%. 

Discount,                           10.56 

2.6394 

(< 

"    2%. 

Proceeds,                        $781.28 

$  10.56     = 

(( 

"    8%. 

(5)  The  same  note.     Discounted, 

November  10, 

at  6%. 

Solution. 

Date  of  maturity,  December  12 

>. 

Discount  term,  Nov.  10  to  Dec 

:.  13  =  33da. 

Amount  due  at  ma- 

$791.84    = 

base  for  discount. 

turity,  as  above,        $791.84 

$    7.9184  = 

discount, 

60  da.,  6%. 

Discount,                             4.36 

3.9592  = 

(( 

30   " 

Proceeds,                       $787.48 

.3959  = 

li 

3   " 

$4.36     = 

tt 

33   " 

ORAL    EXERCISE. 

Find  date  of  maturity. 

Find  time  of  discount. 

Date 

Date  of  Note.          Term. 

Date. 

Term. 

Discount. 

1.      Jan.   27.               2  mo. 

6.     Apr.     9. 

60  da. 

2.      Jan.   31.                1  mo. 

7.     Feb.  17 

1  mo. 

,    Mar.  6 

3.       Jan.   30.             60  da. 

8.      Mar.  12. 

30  da. 

Mar.  26. 

4.       Feb.  28.               4  mo. 

9.     Apr.  19. 

60  da. 

May    7. 

5.       Jan.    30.                6  mo. 

10.      July  14. 

6  mo, 

,    July  29. 

Find  the  discount  and  proceeds  of  these  non-interest  notes  for  full  term 
Face.  Term  of  Note.  Rate  Discount. 

11.  $960  60  da.  6%. 

12.  $600  2  mo.  5%. 

13.  $420  30  da.  6%. 

14.  $840  60  da.  7%. 


322  BUSINESS  ARITHMETIC. 

Face. 


Term  of  Note. 

Rate  Dis. 

4  mo. 

6%. 

30  da. 

6%. 

12  da. 

6%. 

20  da. 

6%. 

Term  of  Note. 

Date  Dis. 

8. 

90  da. 

(6%.)  Mar.  17. 

9. 

90  da. 

(6%.)  July     6. 

15. 

16.  $1200,  discounted  for 

17.  $  840. 

18.  $  920. 

Face.  Date 

19.  $  600  Mar. 

20.  $1200  June  19 

EXERCISE. 

1.  Discount  5%  for  full  term,  the  draft  on  page  310. 

2.  Discount,  at  7%,  for  full  term,  the  draft  on  page  311. 

3.  Discount  at  5%,  for  full  term,  the.  note  on  page  307. 

4.  Discount,  April  14,  at  5%,  the  acceptance  on  page  311. 

5.  Discount  February  19,  1912,  at  6%,  the  note  on  page  316. 

6.  A  merchant  discounts  on  May  16,  at  6%,  the  following  notes  and 
acceptances.     What  sum  should  be  placed  to  the  credit  of  his  account? 

(a)  A  3  mo.  note  for  $725.60,  dated  Apr.  12. 

(6)  A  60  day  sight  draft,  for  $852,  dated  Apr.  20  and  accepted 
Apr.  23. 

(c)  A  4  mo.  note  for  $192.68,  dated  May  16. 

(d)  A  30  day  acceptance,  for  $365.42,  dated  May  15.  i 

ORAL   EXERCISE. 

Find  the  amount  due  at  maturity  on  the  following  notes: 

Face.  Term.  Rate  Int. 

2  mo.  3%. 

60  da.  6%. 

90  da.  4%. 

30  da.  6%. 

4  mo.  6%. 

20  da.  6%. 

90  da.  6%. 

60  da.  5%. 

Find  the  discount  and  proceeds  of  the  following  notes,  bearing  interest 

at  6%  and  discounted  at  the  same  rate: 


1. 

$  246. 

2. 

$    88.: 

3. 

$  900. 

4. 

$1260. 

5. 

$  425. 

6. 

$  960. 

7. 

$  528. 

8. 

$  720. 

LOANS  AND  PAYMENTS.  323 


Face. 

Term. 

Term.  Dia 

9. 

$  600. 

4  mo. 

60  da. 

10. 

$1200. 

60  da. 

20  da. 

11. 

$  900. 

30  da. 

30  da. 

Face. 

Date. 

Term. 

Date  Dis. 

12. 

$  840. 

Jan.     6. 

60  da. 

Feb.    5. 

13. 

$1200. 

June    5. 

30  da. 

June  25. 

14. 

$  600. 

Aug.  10. 

EXERCISE. 

60  da. 

Aug.  30. 

1.  Find  the  discount  and  proceeds  on  the  note  on  page  321 ,  if  discounted 
November  4,  at  6%.     If  discounted  October  27,  at  6%. 

2.  Determine  the  proceeds  of  the  same  note,  if  discounted  at  5%  on 
Nov.  3. 

3.  Find  the  proceeds  of  the  following  firm  note,  if  discounted  March  17, 

at  5%: 

$582.tV7  Kansas  City,  Mo.,  Mar.  5,  1913 

Ninety  days after  date we. , .  .promise  to  pay  to 

the  order  of_ _John  Sheldon 

at our  office 

Five  hundred,    eighty-tvjo y^^Dollars. 

Value  received  with  Interest  at  6%  per  annum. 

J.  M.  Brovmson  &  Bro. 
No.  189        Due 

4.  James  Devlin  is  obliged  to  pay  his  90  day,  5%  interest-bearing  note, 
for  $960,  on  Nov.  19.  He  has  only  $564.30  to  his  credit  in  the  bank 
where  his  note  becomes  payable.  On  Nov.  18,  therefore,  he  discounts 
the  note  shown  below  at  6%.  He  also  deposits  checks  for  $82.67  and  $91.80. 
After  his  own  note  has  been  paid,  what  balance  remains  to  his  credit? 

$468.tTnr  Baltimore,  Md.,  Nov.   18,   190— 

Eighty  days. after  date I promise  to  pay 

to  the  order  of James  Devlin 

at ... .  Third  Nat.  Bank 

. . .  .Four  hundred  sixty-eight xViyDollars 

Value  received  with  Interest  at  5%  per  annum. 

Chas.  D.  Warrenton. 
No.  82        Due.... 


324 


BUSINESS  ARITHMETIC. 


5.     Extend  this  form: 

Discount  Ticket. 

Third  National  Bank. 

March  16,190. 

NOTES   AND    DRAFTS   LEFT   FOR   DISCOUNT. 

By  James  Fielding 


Note  or 
Draft. 


Maker  or  Drawer. 


Face. 


840  — 


1765 

882 

1206 


Int.  to 
Maturity, 


20 

(6%) 
(5%) 


Amount. 


30  d.  n.      S.  P.  Norton 

60  d.  dft.  Johnson  Bros. 

4  mo.  n.     Roberts  and  Bro. 

3  mo.  n.     Champlain  Mfg.  Co.    . 

Total, 

Discount,      @  6%  (full  term) 

Proceeds, 

7.    Complete  the  extension  of  the  bank  discount  register  on  page  325. 
ORAL    EXERCISE. 

1.  If  a  non-interest  note  is  discounted  for  its  full  term  of  4  mo.,  at 
6%,  the  proceeds  equal  what  per  cent,  of  the  face?  If  the  proceeds  are 
$980,  what  is  the  face? 

2.  A  60  day,  6%,  interest-bearing  note  is  discounted  for  30  days  at 
6%.     The  proceeds  equal  what  per  cent,  of  the  face? 

3.  Knowing  the  proceeds  of  a  note,  how  may  its  face  be  determined? 

EXERCISE. 

Note.  The  per  cent,  method,  suggested  above,  may  be  used,  or  the 
proceeds  of  a  dollar  note,  under  the  given  conditions  ,may  be  determined. 
The  proceeds  divided  by  the  proceeds  of  a  dollar  note  will  determine  the 
face. 

1.  What  must  be  the  face  of  a  non-interest-bearing  note  for  3  mo., 
that  will  yield  $750  proceeds  when  discounted  for  its  full  term  at  6%? 

2.  On  January  17, 1  sold  A.  B.  Norton  a  bill  of  merchandise  amount- 
ing to  $762.50,  less  20%.  Terms,  10  days.  At  the  end  of  the  period, 
Norton,  being  temporarily  short  of  cash,  offers  his  non-interest,  30  day 
note  for  such  an  amount  that  when  discounted  for  its  full  term  at  6%,  the 
proceeds  equal  the  face.  This  offer  being  accepted,  the  note  should  be 
made  out  for  what  sum? 

Individual  Original  Work. 
Report  on  local  business  methods  of  computing  discounts  on  notes, 
drafts  and  business  paper.     Give  local  process  of  discounting. 


LOANS  AND  PAYMENTS. 


325 


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CHAPTER   XXXVIII. 

SAVINGS  ACCOUNTS. 

"The  best  way  to  accumulate  money  is  to  resolutely  save  and  bark  a 
fixed  portion  of  your  income,  no  matter  how  small  the  amount,' — The 
WorWs  Work. 

INTRODUCTORY    EXERCISE. 

1.  Name  three  good  reasons  for  saving  money  from  one's  income. 

2.  If  a  journeyman  carpenter  saw  a  chance  to  become  a  contractor 
on  a  small  scale,  he  would  have  what  reason  for  saving  money? 

3.  Direct  savings  increase  rapidly.     A  salesman,   setting  aside  ten 

cents  per  day,  saves  $ per  month.     An  iron-worker,  saving  $3.50  per 

week,  has  $ at  the  end  of  one  year.     It  takes  an  office  clerk,  who  is 

saving  20%  of  his  weekly  salary  of  $12, years  to  save  the  $600  necessary 

to  secure  an  interest  in  a  small  independent  business. 

4.  What  are  the  dangers  of  simply  "hoarding"  one's  money? 

5.  What  difficulties  lie  in  the  way  of  investing  small  sums  as  they  are 
saved? 

6.  What  advantages  result  from  placing  those  small  savings  in  a  bank? 

7.  What  rates  of  interest  do  your  local  banks  and  trust  companies  pay? 

8.  What  is  the  smallest  sum  accepted  by  them  to  open  an  account? 
What  is  the  smallest  sum  accepted  thereafter  as  a  deposit? 

9.  Give  reasons  for  one's  withdrawing  his  deposit  —  (a)  in  prosperous 
times;  (6)  in  times  of  panic;  (c)  in  times  of  business  depression. 

Practically  all  investment  by  wage-earners  starts  in  the 
savings  bank,  an  institution  chartered  by  the  State  for  receiving 
savings  deposits,  on  which  it  pays  interest.  Savings  banks 
receive  enormous  deposits  of  individually  small  amounts. 
A  New  York  bank,  in  a  poor  district,  has  over  $100,000,000  on 
deposit. 

The  savings  banks  usually  have  the  primary  quality  of 
safety,  and  are  more  carefully  regulated  and  guarded  by  law 
than  are  other  banking  institutions.     Most  savings  banks  are 

326 


SAVINGS  ACCOUNTS.  327 

partly  or  entirely  mutual  and  are  run,  in  such  cases,  for  the 
benefit  of  the  depositors.  A  few  are  stock  companies.  Unless 
run  for  the  benefit  of  stock  holders,  the  banks  place  the  major 
portion  of  their  funds  in  safe  mortgages,  or  in  government, 
or  other  "gilt  edge"  bonds.  Often  the  state  laws  specify  very 
definitely  the  class  of  investments  that  may  be  made.  It  is 
well  to  examine  the  laws,  or  to  get  reliable  business  advice 
before  depositing  in  any  bank.  If  good  banks  are  not  within 
easy  reach,  deposits  may  be  sent  to  them  by  mail. 

Many  trust  companies  also  accept  savings  deposits.  They 
generally  hold  deposits  '^  subject  to  check,"  but  are  not 
always  as  closely  safeguarded  as  savings  banks. 

EXERCISE. 

1.  Eleven  savings  banks  of  Pennsylvania  recently  had  398,885  de- 
positors, an  average  of  ?  per  bank. 

2.  Complete  this  tabulation  concerning  the  savings  banks  of  the 
country : 

Year.    No.  Banks.       Deposits.        Depositors.  Av.  per  Bank.  Av.  Depos. 

1905    '     1237  $3,093,077,357   7,693,229  $ $ 

1910         1759.  4,070,486,247  9,142,908  $ $ 

5  yr.  increase  ?  ?  ?  ? 

%  of  increase  ?  ?  ?  ? 

3.  Account,  in  some  way,  for  these  increases. 

Most  deposits  are  for  long  periods.  Withdrawals  are  at 
infrequent  intervals,  and  are  further  restricted  by  the  fact 
that  a  bank  may  require,  if  it  desires,  10  days  to  60  days  notice 
before  making  a  payment.  This  rule  is  seldom  enforced  except 
in  times  of  money  stringency  or  sudden  panic.  On  notice 
of  expected  withdrawal,  the  bank  begins  to  save  incoming 
deposits  to  meet  this  sum.  It  thus  leaves  practically  its  entire 
savings  receipts  free  for  investments.  Money  that  it  cannot 
immediately  invest,  it  may  lend  "on  call"  (see  p.  345),  or  lend 
to  commercial  banks  at  1%  to  2%  interest. 

Whether   mutual   or   stock   organizations,    savings   banks 


328 


BUSINESS  ARITHMETIC. 


usually  pay  from  3%  to  5%,  the  average  for  the  country 
being  3f  %.  Interest  is  added  to  deposits  at  periodic  in- 
tervals. Since  the  period  of  compounding  has  some  effect 
on  income,  it  is  well  to  note  this,  as  well  as  the  method  of 
computing  interest,  before  selecting  a  bank.  The  method  is 
generally  stated  in  the  rules  of  the  bank. 

EXERCISE. 

(Use  compound  interest  table,  page  303.) 
Find  the  amount  for  one  year: 


Of  $400  compounded  semi-annually  at  4%. 

Of  the  same  sum  compounded  annually. 

Of  $364  compounded  quarterly  at  4%. 

Of  $942  compounded  semi-annually  at  3|%. 

Find  the  amount  of  $368  for  three  years  if  compoimded  annually 


1. 
2. 
3. 
4. 
5. 
at  3%. 

Since,  in  most  cases,  the  amount  on  deposit  is  constantly 
changing,  a  special  method  is  necessary  for  computing  the 
simple  interest  for  each  term.  Many  savings  banks  allow 
interest  on  the  smallest  balance  during  any  interest  period. 
Sums  deposited  early  in  a  period  draw  no  interest  until  the 
next  period,  unless  deposited  the  first  day  of  the  period. 
Ffactions  of  a  dollar  do  not  draw  interest. 

ILLUSTRATION  I. 
Northern  Savings  Bank. 
In  account  with  James  P.  Norman. 


Date. 

Withdrawn. 

Deposited. 

Interest. 

Balance. 

190— 

1 

Jan. 

1 

29 

10 

320 
310 

Feb. 

17 

462 

772 

Mar. 

2 

90 

50 

1 

68150 

Apr. 

1 

3,10 

684 

60 

4 

40 

724 

60 

June 

28 

160 

564 

60 

July 

1 

5164 

570 

24 

SAVINGS  ACCOUNTS. 


329 


Explanation.  The  deposits  and  withdrawals  are  entered  in  order  of 
dates.  The  rate  in  this  bank  is  4%,  compounded  quarterly,  or  1%  per 
quarter.  The  first  interest  period  extends  from  Jan.  1  to  March  31.  The 
smallest  balance  on  hand  during  the  period  is  S3 10,  from  Jan.  29  to  Feb.  17. 
The  interest  is  1%  of  $310,  or  $3.10,  which  is  added  to  the  balance,  as  a 
new  deposit,  on  Apr.  1.  The  next  interest  period  extends  from  Apr.  1 
to  June  30.  The  smallest  balance  is  $564.60,  on  June  28,  and  the  interest 
is  1%  of  that  balance,  or  $5.64.     This  interest  is  added  July  1. 

If  compounded  semi-annually,  the  smallest  balance  for  the  entire  period 
is  $310,  and  the  interest  is  2%,  or  $6.20,  a  loss  to  the  depositor,  by  change 
of  compounding,  of  $2.54. 

Some  savings  banks  reckon  interest  on  monthly  or  quarterly 
balances,  but  compound  it,  or  add  it  to  the  deposit,  only  at 
semi-annual,  or  annual,  intervals. 

ILLUSTRATION   II. 

Washington  Savings  Bank. 
In  account  with  Henry  C.  Brewster. 


Date. 

Withdrawn. 

Deposits. 

Interest. 

Balances. 

191— 

July 

1 

560 

Aug. 

5 

246 

10 

806 

10 

Sept. 

8 

26 

780 

10 

19 

129 

80 

909 

90 

Oct. 

31 

40 

869 

90 

Dec. 

14 

37 

43 

907 

33 

Jan. 

1 

14 

29 

921 

62 

Explanation.  Here  assume  interest  at  4%,  on  quarterly  balances, 
compounded  semi-annually.  The  lowest  balance  in  the  quarter  ending 
Sept.  30  is  $560,  and  the  interest  is  1%,  or  $5.60.  The  lowest  balance  in 
the  fourth  quarter,  ending  Dec.  31,  is  $869.90  and  the  interest  is  $8.69. 
The  total  interest,  compounded  on  Jan.  1,  is  $5.60-}- $8.69,  or  $14.29. 

EXERCISE. 

1.  What  is  the  effect  of  a  heavy  withdrawal  just  before  the  end  of  an 
interest  period? 

2.  Would  Norman  (illus.  1)  have  gained  by  delaying  his  withdrawal 
of  June  28,  until  July  1? 

3.  When  would  an  additional  withdrawal  of  $400  have  caused  the 
greatest  "interest"  loss  in  Brewster's  account? 


330 


BUSINESS  ARITHMETIC. 


4.  Determine  the  interest  and  final  balance  of  Brewster's  account,  if 
the  interest  is  reckoned  on  monthly  balances. 

5.  Balance    this    account.     Rate    3%,    reckoned    and    compounded 
quarterly : 

Thornton  Savings  Bank. 
In  account  with  John  Kensington. 


Date. 

Withdrawals. 

Deposits. 

Interest. 

Balances. 

190— 

Jan. 

1 

8 

31 

40 

72 

50 

362 

40 

Mar. 

3 

80 

45 

May 

19 

162 

65 

June 

12 
30 

40 

321 

6.    Balance  this  account.     Rate,  4%,  reckoned  on  quarterly  balances, 
and  compounded  semi-annually. 

Illinois  Center  Savings  Bank. 
In  account  with  C  R.  Martin. 


Date. 

Withdrawn. 

Deposited. 

Interest. 

Balances. 

July 

5 
29 

102 

10 

328 

45 

Aug. 

15 

107 

38 

Sept. 

29 

54 

13 

Nov. 

10 

28 

127 

69 

24 

63 

Dec. 

12- 
18 

132 

46 

31 
50 

7.    Extend  this  account.     Rate,  4%,  reckoned  on  monthly  balances 
and  compounded  semi-annually. 

Exchange  Savings  Bank. 
In  account  with  Robert  S.  Fenwick. 


Date. 

Deposits. 

Interest. 

Payments. 

Balance. 

Jan. 

1 

582 

61 

Mar. 

3 

45 

May 

16 

312 

62 

June 

21 

46 

58 

Sept. 

17 

92 

46 

Oct. 

20 

545 

20 

Nov. 

10 

416 

92 

Dec. 

21 

38 

80 

SAVINGS  ACCOUNTS. 


331 


Note.     Notice  change  in  column  headings. 

8.     Balance  this  "subject  to  check"  trust  company  account.     Interest, 
2%,  compounded  semi-annually.     $240  is  on  hand  January  1,  1912. 

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CHAPTER  XXXIX. 

STOCKS  AND   BONDS. 

The  proprietor  of  a  business  who  finds  a  need  for  increased 
capital  may  secure  it  by  taking  into  partnership  the  one  who 
suppHes  it.  But  partnerships  are  often  unsatisfactory,  because 
each  partner  is  usually  responsible  for  the  acts  of  the  others 
and  in  most  cases  is  liable  for  all  debts.  Each  change  of 
partners,  moreover,  causes  a  new  adjustment  of  interests  and 
often  interrupts  business.  These  objections  increase  with  the 
enlargement  of  a  business  and  often  seriously  affect  ease  and 
continuity  of  management. 

The  corporation  meets  many  of  the  objections  to  a  partner- 
ship. It  is  an  association  of  individuals  recognized  under  the 
law  as  a  single  artificial  person.  The  capital,  or  capital  stock, 
of  such  an  organization  is  divided  into  small  equal  amounts 
called  shares.  A  single  person  may  hold  one  or  more  shares. 
Many  individuals  may  be  interested  in  large  companies. 
A  single,  well-known  railway  corporation  has  63,000  share- 
holders or  stockholders. 

The  steps  in  the  organization  of  a  corporation  are: 

(1)  A  specified  number  of  people  (at  least  three,  in  New  York)  come 
together  and  draw  up  a  certificate,  stating  (a)  the  name  of  the  proposed 
corporation,  usually  including  the  word  "Company";  (6)  the  place  of 
business;  (c)  the  nature  of  business;  {d)  the  amount  and  face  (par)  value  of 
shares;  (e)  duration  of  corporation,  (/)  other  information  required  by  law. 

(2)  This  certificate  is  filed  with  the  Secretary  of  State,  who  issues  a 
license  to  these  people  to  open  books  for  subscriptions  to  the  capital  stock. 

(3)  When  the  legal  amount  of  subscriptions  have  been  made,  and  in 
some  instances  paid  in,  a  meeting  of  subscribers  is  called,  officers  are  chosen 
and  by-laws  adopted.  Usually  the  stockholders  elect  directors,  each 
stockholder  Having  as  many  votes  as  he  holds  shares,  and  the  directors 
elect  officers. 

332 


STOCKS  AND  BONDS.  333 

(4)  A  record  of  these  proceedings  is  filed  with  the  Secretary  of  State 
who  then  issues  a  certificate  to  the  effect  that  the  corporation  is  legally 
organized  and  ready  for  business.  ^ 

Certificates  of  stock  are  issued  to  subscribers  when  sub- 
scriptions are  paid,  and  these  certificates  may  be  transferred 
at  will  thereafter,  unless  the  stockholder  is  indebted  to  the 
corporation.  In  such  cases,  transfer  may  be  permitted  by 
special  vote. 

Advantages.  The  principal  advantages  of  corporations  to 
their  partners  (stockholders)  are:  (1)  Financial  responsibility 
for  only  the  par  value  of  stock  (twice  the  value  in  bank  stock) 
until  paid;  (2)  small  shares,  permitting  the  purchase  of  interests 
in  many  prosperous  businesses  where  a  partnership  interest 
could  not  have  been  bought;  (3)  a  simple  method  of  transfer 
of  ownership ;  (4)  possibility  of  holding  an  interest  in  a  business 
for  which  one  has  neither  time  nor  qualifications  for  direct 
control;  (5)  the  value  of  stock  as  an  investment  for  surplus 
funds.  The  advantages  to  the  business  are  (1)  possibility  of 
indefinite  expansion  as  to  capital  and  organization;  (2)  unity 
of  direction  and  management,  enabling  the  employment  of 
experts. 

Shares.  The  par,  or  face  value  of  shares  in  the  larger 
corporations  is  usually  $100,  but  may  be  any  simple  multiple 
or  ahquot  part  of  $100,  as  $1,  $5,  $10,  $500,  etc.  A  majority 
of  shares  outstanding  control  the  company.  Thus,  a  few  large 
shareholders  may  control  a  company  in  which  a  host  of  small 
holders  are  interested. 

EXERCISE. 

Find  the  missing  values: 

Capital.  Par.  Val.  per  Sh.                        No.  Shares. 

1.  $    200,000  $100                                          

2.  $4,000,000  $  50                                          

3.  $      80,000  $  10                                           

4.  $    250,000                                  500 


334  BUSINESS  ARITHMETIC. 

5.  $    600,000  $200  

6.    $400  1500 

7.  Andrews  holds  1200  shares  ($100)  m  a  $250,000  company.  He 
must  buy shares,  in  order  to  control  it. 

8.  Name  the  number  of  shares  necessary  to  control  each  of  the  com- 
panies in  ex.  1-7. 

9.  In  the  reorganization  of  a  $50,000  company,  the  par  value  of  shares 
is  changed  from  $20  to  $100.  Snell,  who  owns  400  $20  shares,  should 
receive $100  shares. 

Preferred  and  Common  Stock.  Usually  the  original  issue 
of  stock  is  called  common  stock  and  has  full  voting  power. 
Sometimes  ^preferred  stock  is  issued  at  time  of  organization 
and  may  or  may  not  have  voting  power.  Preferred  stock- 
holders must  be  paid  a  certain  stated  proportion  of  profits 
before  common  stockholders  receive  anything.  Thus,  "4% 
preferred  stock"  entitles  the  holders  to  4%  of  the  face  value  in 
profits,  annually,  hut  no  more.  The  preferred  stock  may  be 
cumulative,  carrying  over  to  a  succeeding  year  its  right  to  its 
share  for  a  previous  year,  even  if  in  those  years  there  was  no 
profit.     Preferred  stock  is  sometimes  non-cumulative. 

Preferred  stock  is  issued,  sometimes,  to  meet  special  needs 
for  additional  capital,  and  is  in  the  nature  of  a  perpetual  loan, 
the  holders  having  no  right  to  vote.  If  the  common  and 
preferred  stockholders  agree,  a  second  issue  of  preferred  stock 
may  be  made  that  will  have  rights  over  previous  issues. 

Raising  the  Capital.  Some  states  require  that  the  original 
capital  stock  shall  be  paid  in  full;  in  other  states  it  may  be 
sold  for  what  it  will  bring.  The  market  price  is  termed  a 
quotation.  If  the  stock  is  sold  at  face  value,  quotation  is 
"at  par";  if  below,  it  is  ''below  par,"  or  at  a  discount;  if  in 
advance  of  par  it  is  "above  par,"  or  at  a  premium. 

At  times,  private  bankers  will  take  over  a  large  block  of 
stock,  not  originally  subscribed  for,  at  a  fixed  price  and  sell  it 
to  the  public.     Often  a  commission  is  paid,  in  money  or  stock. 


STOCKS  AND  BONDS.  '  335 

EXERCISE. 
Find  missing  values. 

Actual 

Capital  Stock.      Shares.         Orig.  Selling  Quotation.  Cash  Cap. 

1.  $200,000  $100  $100  $ 

2.  $500,000  $100  $98  $ 

3.  $2,000,000  $50  $47  $ 

4.  $500,000  $100  $103  $ 


5.  C.  P.  Jacobs  has  subscribed  for  500  $100  shares  in  a  new  company, 
payable  in  full.  He  turns  over  a  small  factory,  which  is  accepted  at  a 
valuation  of  $36,492.     What  must  he  pay  in  cash? 

6.  The  partnership  firm  of  Adams  and  White  is  reorganized  is  a  cor- 
poration of  $60,000  capital,  in  $100  shares.  Adams'  present  worth  in 
the  old  firm  was  $18,500  and  W^hite's  $32,000.  What  number  of  shares 
should  each  receive  for  his  present  w  orth,  and  what  nuinber  of  shares  should 
go  into  the  treasury  for  future  sales? 

7.  Suppose  the  full  number  of  shares  is  divided  between  Adams  and 
White  in  proportion  to  their  present  worth,  Adams  paying  White  in  cash 
for  the  fraction  of  a  share,  what  number  of  shares  should  each  receive? 
The  capital  stock  is  watered,  or  in  excess  of  the  real  capital  by  $ . 

8.  Martin  agrees  to  take  80  shares  ($100),  Brown  450  and  Baker  the 
balance,  in  an  $80,000  company.  They  decide  to  pay  for  their  stock  at 
15%  premium,  in  order  to  create  a  surplus  or  extra  working  capital. 
Determine  the  capital,  surplus  and  payment  of  each  subscriber. 

Distribution  of  Profits  and  Losses.  In  case  of  losses,  if  stock 
is  assessable,  stockholders,  by  vote  of  directors,  may  be  re- 
quired to  pay  a  certain  assessment,  levied  as  an  amount  per 
share,  or  as  a  certain  per  cent,  of  par  value.  The  profits, 
on  the  other  hand,  are  held  as  undivided  profits  or  surplus, 
until  dimdends  are  declared  by  the  directors.  Dividends  are 
more  commonly  expressed  in  dollars  per  share,  or  as  a  per 
cent,  of  par  value  of  stock.  Dividends  must  be  paid,  first,  on 
preferred  stock,  and  then  on  common  stock.  The  directors 
are  not  forced  to  dispose  of  all  profits  as  dividends. 

ORAL    EXERCISE. 

1.  The  Crown  Manufacturing  Co.,  capitaHzed  at  $1,000,000  ($100 
shares)  declares  a  dividend  of  5%  ($5  per  share).     The  total  dividend 


336  BUSINESS  ARITHMETIC. 

in  money,  is  $ ;  and  the  dividend  of  Jamison  who  owns  60  shares  is 

60X$5,  or$ . 

Find  the  missing  values. 

Capital.    No.  Sh.   Par  Val.  Div.  % 

2.  $200,000  1000     ?     4% 

3.  $500,000   ?     $100    5% 

4.  $80,000   ?      $50     ? 

5.  $2,400,000        ?  $100  ? 

6.  $60,000       ?  ?  4i% 

EXERCISE. 

Note.     Assume  par  value  as  $100  per  share,  unless  otherwise  stated. 

1.  What  sum  of  money  is  required  to  pay  a  3j%  dividend  on  a  com- 
pany capitaUzed  at  $4,000,000? 

2.  A  7%  dividend  on  670  shares  of  P.  R.  R.  stock  will  yield  the  owner 


3.  A  quarterly  dividend  rate  of  1|%  will  yield  the  owner  of  250  shares 
$ per  year. 

4.  What  rate  of  dividend,  in  dollars  per  share,  must  be  declared  in 
order  to  distribute  $42,000  to  stockholders  in  a  $500,000  company? 

5.  An  assessment  to  meet  a  loss  of  $36,000  must  be  levied  against  a 
capital  stock  of  $400,000  ($50  shares).  Determine  the  assessment  rate 
A  must  pay  $ on  his  holdings  of  120  shares. 

6.  An  industrial  corporation,  capitalized  at  $2,000,000,  has  net  earnings 
of  $172,  541.50,  for  the  year.  Of  this  sum,  20%  is  transferred  to  surplus. 
What  is  the  largest  whole  per  cent,  of  dividend  that  can  be  declared  from 
the  balance?  If  this  is  declared  and  paid,  what  is  the  balance  of  undivided 
profits? 

7.  The  Concklin  Wheel  Co.  has  a  capital  of  $200,000,  one-fourth  con- 
sisting of  5%  preferred  stock.  Its  gross  income  for  one  year  is  $729,342, 
and  its  gross  expense  $512,860.  The  directors  set  aside  10%  of  profits 
for  surplus,  and  after  paying  preferred  dividends,  declare  the  largest  possible 
dividend  (in  per  cents  and  fourths).     The  rate  declared  is %. 

Note.  At  times,  when  profits  are  very  high  or  when,  for  any  reason,  a 
cash  payment  is  not  advisable,  stock  dividends  are  sometimes  declared. 
Additional  stock,  thus  issued,  participates  in  all  future  dividends.  When 
business  is  profitable,  but  there  is  a  temporary  money  stringency,  dividends 
are  sometimes  paid  in  scrip,  or  certificates  cashable  after  a  certain  date. 

8.  The  Lamberton  Electric  Co  ,  capitalized  at  $500,000,  has  decided 
to  distribute  $80,000  in  dividends.     The  highest  rate  it  pays  in  cash  is 


STOCKS  AND  BONDS.  337 

8%,  the  balance  being  paid  in  new  stock.     Harper,  who  owns  400  shares, 

receives  $ cash  and new  shares.     The  new  capital  stock  of  the 

company  is  $ . 

Bond  Issues.  When  a  government  or  corporation  borrows 
large  sums  of  money  it  usually  issues  a  formal  promissory  note 
called  a  bond.  The  bonds  usually  are  issued  in  series  of  like 
tenor  and  for  equal  amounts,  with  interest  payable  annually, 
semi-annually  or  quarterly.  The  face  value  is  usually  $1000, 
but  $500  bonds  are  common,  and  smaller  ones  are  issued. 

The  payment  of  industrial  bonds  is  secured  by  a  mortgage 
on  the  property  of  the  corporation.  Minor  municipal  bonds 
are  sometimes  paid  from  a  sinking  fund  raised  by  taxation. 
Government  bonds  and  securities,  and  corporation  debenture 
bonds  have  no  security.  First  mortgage  bonds  usually  have  a 
first  right  to  lien  on  property;  succeeding  issues,  second,  third, 
etc.,  have  rights  only  after  the  first  issue  is  paid. 

Bonds  are  issued  commonly  in  registered  or  coupon  form. 
If  registered,  the  names  of  the  owners  are  recorded  in  the  books 
of  the  company,  as  in  the  case  of  stock;  the  bonds  are  made 
payable  to  the  specified  buyer,  and  are  transferable  by  endorse- 
ment and  office  record.  Interest  is  sent  by  check  to  the 
recorded  owner.  Coupon  bonds  have  attached  coupons  or 
vouchers,  one  for  each  interest  period  during  the  life  of  the 
bond.  These  are  detached  as  they  become  due,  and  are 
presented  for  payment,  or  collected  through  a  bank. 

Many  bonds  derive  their  name  from  their  rate  of  interest, 
and  from  their  term  and  time  of  maturity.  "  D.  C.  4s  of  1925  " 
are  District  of  Columbia  4%  bonds  due  in  1925. 

While  bonds  are  usually  payable  at  a  fixed  time,  provision 
is  sometimes  made  to  pay  off  a  certain  number  of  an  issue  each 
year.  Sometimes  the  company  reserves  the  option  of  paying 
off  between  certain  dates,  as  "after  ten  years  but  not  later 
than  twenty  years." 

Bonds  are  similar  to  preferred  stock  in  that  they  pay  a  fixed 
23 


338  BUSINESS  ARITHMETIC. 

rate  of  dividend  or  interest.  They  have  a  Hmited  life  and 
they  confer  no  voting  power,  but  they  have  a  right  to  the 
payment  of  interest  ahead  of  the  preferred  stock. 

EXERCISE. 

1.  The  Selford  Electric  Co.  sells  a  $40,000  6%  bond  issue  at  $1050 
per  $1000  bond,  receiving  $— .  The  interest  on  the  issue  will  cost  the  com- 
pany $  —  annually. 

2.  The  Crane  Hardware  Co.  issues  5%  bonds  running  five  years  for  a 
face  value  of  $120,000.  The  company  creates  a  4%  sinking  fund  to  meet 
the  issue  at  maturity.  If  it  sets  aside  equal  amounts  at  the  end  of  each 
year  and  also  pays  interest,  what  is  the  annual  expense? 

.  3.  The  Western  Banking  Corporation  underwrites  the  bonds  of  the 
West  River  Lumber  Co.,  par  value  $50,000,  at  90%  of  face  value.  The 
banking  corporation  disposes  of  the  issue  at  93%  of  face  value,  thus  making 
a  profit  of  $  — ,  and  paying  the  lumber  company  $  —  . 

Buying  and  Selling.  The  market  quotations  are  affected 
by  many  varied  factors.  Primarily,  the  price  should  depend 
on  earning  power,  present  and  prospective.  Thus,  price  is 
influenced  indirectly  by  the  capital  of  the  company,  for  over- 
capitalization means  les^  income  per  share.  The  price  may 
be  seriously,  though  temporarily,  influenced  by  speculation, 
or  by  an  attempt  of  parties  to  gain  control  of  a  company.  The 
general  prosperity  or  depression  of  industry  affects  quotations 
seriously.  A  financial  depression  may  cause  liquidation  or  the 
selling  of  stock  to  meet  financial  debts  —  sales  often  made  at 
a  heavy  loss.  A  sudden  loss  of  confidence  in  one  stock  may 
affect  others.  The  fear  of  injurious  legislation,  the  threat 
of  war,  the  falling  off  of  exports,  the  closing  down  of  a  great 
factory  are  other  illustrations  of  the  many  influences  affecting 
market  price. 

Quotations.  General  purchases  and  sales  are  governed 
largely,  as  to  price,  by  the  quotations  of  the  stock  exchanges. 
The  quotation  lists,  as  printed,  often  vary  in  form,  but  usually 
show  the  price  in  dollars  per  share,  at  several  times  of  the 


STOCKS  AND  BONDS.  339 

business  day.     For  bonds,  the  price  is  quoted  per  $100  of  par 
value. 

Stock  Quotation  List. 
(From  newspaper  lists.     Consult  newspapers  for  current  quotations.) 

Open.  High.  Low.  Close. 

Mo.,  Kan.  &  T.,  com 26  26  25^       25^ 

M.S.  P.  &S.  S.  M 1411  142       141       142 

Missouri  Pacific 44  44^  43|      43| 

N.  Y.  Air  Brake 82f  82f  82t  .  82f       • 

New  York  Central 109^  110  107  107i 

N.  Y.,  Ont.  &  West 33  33^  32f      32^ 

Norfolk  &  Western 113  113  113  113 

Northern  Pacific 126|  127|  125 f  126 

Pennsylvania  R.  R 124  124i  123i  123^ 

Pressed  Steel  Car 35  35  35        35 

Rail's  Steel  Springs 36  36  35        35 

Reading 164^  165^  162|  163J- 

Rep.  S.  &  I.,  pfd 89  89  89        89 

Rock  Island,  com 24i  24^  24         24 

Rock  Island,  pfd 47|  47|  47|      47f 

Bonds. 

C.  R.  I.  &  P.  4s 69        69  68^  68| 

Atcn.  4s.  con 104^  104^  104^  104| 

Nor.  &  West  48 98^      98  98^  98f 

U.S.  Steel  5s 101|  102  lOlf  102 

Government  Bonds. 

United  States  38,  coupon 104f  104f  104|  104f  5 

United  States  4s,  coupon 114^  114^  114^  114^  15 

Imp.  Japanese  Gov.  6s  ctfs.  full  paid.  .  90f  92  91  f  92  678 

Republic  of  Cuba  5s,  full  m.  paid 103  103  102^  102 1  232 

In  the  above  stock  list,  N.  Y.  C.  stock,  for  example,  is 
quoted  at  the  opening  of  the  day  at  $109.50.  During  the  day, 
the  highest  price  was  $110  and  the  lowest,  $107.  At  the  close 
of  the  day,  the  price  was  $107,125.  Such  stock  is  usually 
bought  or  sold  through  brokers.  The  common  charge  is 
1/8%  of  par  value,  or  $1/8  on  a  $100  share.  Small  brokers, 
handling  small  lots,  sometimes  charge  1/4%. 


340  BUSINESS  ARITHMETIC. 

Illustrations.  (1)  Determine  the  cost  of  300  shares  of  P.  R.  R.  stock, 
at  closing  quotation,  brokerage  1/8. 

Solution.  Bj'^  reference  to  list,  the  closing  quotation  is  $123.25.  Broker- 
age is  $1/8  or  $.125. 

The  entire  cost  of  1  share  is  $123.25  +  $.125,  or  $123,375.  300  shares 
cost  300  X  $123,375,  or  $37,012.50. 

(2)  What  should  I  receive  from  the  sale  of  400  shares  Reading  at  opening 
quotation,  brokerage  1/8? 

Solution.     Opening  quotation,  1641. 

Net  proceeds  of    1    share,  $164.  |  -  $|  =  $164f. 
Net  proceeds  of  400  shares  =  400  X  $164 1  =  $65,750. 

ORAL    EXERCISE. 

1.  Which  of  the  stocks  in  the  quotation  list  are  jyref erred? 

2.  Was  the  general  tendency  of  prices,  during  the  day,  upward  or 
downward? 

3.  If  I  buy  Reading  at  highest  quotation,  brokerage  1/8,  I  pay  $  — 
per  share.  If  I  had  sold  at  the  same  price  I  would  have  received  $  —  net 
proceeds. 

4.  Brokerage  being  1/8,  determine  the  cost  of 

No.  shares.  Stock  or  Bond.  Quotation, 

(a)       100  N.Y.  Air  Brake  Closing. 

•      (h)     1000  Mo.  Pac.  Open. 

(c)  10  Penn.  R.  R.  High. 

(d)  2  No.  Pac.  Low. 
(c)  $10,000,  par.  val.  Nor.  &  Western  High. 
(/)     $10,000,  par  val.     U.  S.  4s,                                 Open. 

5.  Determine  the  net  proceeds,  brok.  1/8,  from  the  sale  of 

Shares  or  Val.  Stock  or  Bond.  Quotation 

(a)     200  sh.  U.  S.  Steel,  68S. 

(6)     $  1,000.  Southern  Pacific,  106f. 

(c)  $10,000.  S.  L.  &  S.  F.,  com.,  16. 

(d)  $10,000.  U.  S.  ?s.,  Closing. 

(e)  100  sh.  Reading,  Closing. 

6.  Determine  the  brokerage  at  1/4  on  lots  of:  80  sh.,  25  sh.,  1000  sh., 
250  sh. 

7.  If  brokerage  is  1/8,  one  way  only,  find  the  net  profit  or  loss  per  share 
on: 

Buying  Quot.     Selling  Quot.  Buying  Quot.   Selling  Quot. 

(a)         79i  81  i  (c)     8    prem.  6  prem. 

(Jb)       lllf  112  id)    51-  dis.  1  prem. 

8.  Express  these  discounts  in  other  forms:  105  (Ans.  5  prem.  or  5 
above  par),  96,  110^,  8  dis.,  7 J  prem.;  5^  below  par,  100. 


STOCKS  AND  BONDS.  341 


EXERCISE. 


1.  Determine  the  .cost  at  opening  quotation,  brokerage  1/8,  of  200  sh. 
M.K.&T.com.;  350sh.  Mo.  Pac;  3000  sh.  Reading;  $10,000  par  of  C, 
R.  I.  &  P.  4s;  $20,000  par  U.  S.  Steel  5s. 

2.  Determine  the  net  proceeds  received  from  the  sale,  at  high  quo- 
tation, brokerage  1/8,  of:  3000  sh.  N.  Y.  C;  2000  sh.  Nor.  &  West.;  800  sh. 
Rock  Is.  com. ;  3000  sh.  P.  R.  R. ;  18,000  sh.  Pressed  Steel  Car;  $500  Japanese 
Gov't  6s. 

Note.  In  determining  the  gain  or  loss  from  a  purchase  or  sale,  it  is 
advisable  to  find  the  net  gain  or  loss  per  share,  and  to  multiply  by  the 
number  of  shares. 

Illustration.     Determine  the  profit  or  loss  from   800  sh.  bought  at 
98f  and  sold  at  101,  brok.  |  each  way. 
Solutwn.     The  cost  =  98|  +  i  =  98|. 

The  net  proceeds    =  101  -  i  =  100 1. 
The  net  profit  per  sh.  =  100 1-  -  98|  =  2f. 
The  total  profit  =  800  X  $2f  =  $2100. 

3.  Find  the  profit  or  loss  in  the  following,  brokerage  \  one  way: 


No.  Sh.  or  Val. 

Stock. 

Buy.  Quot. 

Sell.  Quot. 

(a)      200  sh. 

Penn.  R.  R. 

Open 

1131 

(6)  $8000 

Japanese  6s. 

High 

Close. 

(c)      500  sh. 

No.  &  West. 

Open 

Close. 

{d)    1200  sh. 

No.  Pac. 

Open 

Close. 

(e)    3000  sh. 

Reading. 

1041 

Close. 

(J)  On  $5000  of  each  of  the  bonds  in  the  quotation  list,  buying  at 
opening  and  selling  at  closing  price. 

4.  I  directed  my  broker  to  sell  $20,000  U.  S.  Steel  5s  and  invest  proceeds 
in  Reading.  Buying  quotation,  104 1;  selling  quotation,  88 1;  brokerage 
1/8  each  way.     Number  of  shades  bought?     Surplus? 

Note.  No  fraction  of  a  share  is  bought.  In  purchases  through 
exchanges,  lots  of  hundreds  of  shares  are  usually  dealt  in. 

5.  I  ordered  my  broker  to  sell  2600.  shares  N.  Y.  Ont.  &  W.  (32|) 
and  to  buy  2000  shares  N.  P.  (125 f).  He  should  draw  on  me  for  what 
sum  to  settle  the  balance? 

Find  missing  values: 

6.  Carroll  County  issues  ten-year  coupon  school  bonds,  bearing  6% 
interest,  payable  semi-annually,  which  are  disposed  of  at  102^,  brok.  \. 

The  bonds  produce  a  fund  of  $ ,  but  the  County  must  meet  interest 

charges  of  $ every  six  months,  or  $ in  ten  years.     At  the  close  of 

the  final  year  it  must  pay  out  $ .     The  bonds  have  coupons 

attached,  each  worth  $ ,  if  the  face  value  is  $500  each.     Abbott,  who 


342  BUSINESS  ARITHMETIC. 

buys  $12,000  worth,  pays  $ ,  and  gets  an  annual  income  of  $ ,  or 

— %  on  his  original  investment. 

Stock  Exchanges.  Stock  exchanges  are  organizations, 
formed  for  the  purpose  of  supplying  a  regulated  market  for 
the  purchase  and  sale  of  stocks  and  bonds.  All  cities  of  any 
size  have  such  organizations,  but  the  New  York  Stock  Ex- 
change is  by  far  the  best  known,  and  is  national  in  scope. 
This  has  1000  members.  Membership  is  called  a  seat,  and 
may  be  sold.  The  buyer  of  a  seat  must  be  elected  by  the 
Committee  on  Admissions. 

The  strictest  regulations  are  in  force  in  regard  to  methods 
of  doing  business,  and  the  organization  maintains  the  highest 
standard  of  commercial  integrity  among  its  members.  Trans- 
actions involving  hundreds  of  thousands  of  dollars  are  con- 
tracted verbally  and  are  carried  out  to  the  letter.  Severe 
penalties  are  inflicted  for  any  breach  of  rules. 

An  Exchange  handles  listed  and  unlisted  securities.  It 
does  not  guarantee  the  value  of  securities  that  it  admits  or 
lists,  but  before  listing,  it  makes  examinations  of  the  com- 
panies whose  stock  is  offered  for  listing.  Railway  companies, 
for  example,  must  file  statements  giving  location  and  detailed 
description  of  property,  and  a  financial  statement  of  liabihties, 
assets,  earnings,  mortgages,  etc.  An  industrial  corporation 
must  prove  that  it  is  legally  organized,  and  must  also  make  a 
statement  of  its  financial  and  busii  ^ss  affairs.  An  Exchange 
makes  further  requirements  in  regard  to  regular  reports,  notices 
to  stockholders,  etc.,  thus  forcing  the  company  to  make  public 
its  actual  condition,  before  listing  is  granted. 

For  corporations  of  standing  that  are  unable,  or  unwilling, 
for  some  reason,  to  conform  to  the  strict  requirements  of 
"listing,''  an  "unlisted  department"  is  made.  Less  strict 
statements  are  required,  but  the  stocks  are  dealt  in  in  prac- 
tically the  same  way.  Unlisted  securities  are  not  considered 
of  so  high  a  rank  as  listed  securities,  and  are  not  so  highly 
esteemed  by  bankers  as  collateral  for  loans. 


STOCKS  AND  BONDS.  343 

On  one  Exchange  business  is  done  by  members  for  outsiders 
at  1/8%  commission;  by  members  for  named  members,  at 
1/50%  and  by  members  for  unnamed  members  at  1/32%. 
Lots  of  100  shares,  or  $10,000  par  of  bonds,  are  usually  dealt  in. 
Trading  must  be  done  between  10:00  A.M.  and  3:00  P.M. 
Brokers  often  meet  ''on  the  curb,"  or  in  the  street  in  the 
neighborhood  of  the  exchange,  and  buy  or  sell  small  lots, 
unlisted  securities,  etc.     This  is  the  ''  Curb  Market." 

All  transactions  are  by  word  of  mouth.  There  are  posts  in 
places  in  the  Hall  of  the  Exchange  where  specified  stocks  are 
traded  in.  Brokers  desiring  to  buy  or  sell  the  given  security 
congregate  at  these  posts  and  shout  their  orders  or  sales.  A 
nod,  a  sign  with  the  finger,  or  the  word  ''  sold,"  may  be  the 
sign  of  a  transaction  involving  thousands  of  dollars.  Memo- 
randa are  roughly  made  on  paper,  at  the  first  possible  moment, 
and  later  compared.  Prices  rise  or  fall  by  1/8%.  If  a  broker 
bids  98  for  a  stock  and  no  one  offers  to  sell,  his  next  higher  bid 
is  98|.  During  the  times  a  corporation's  books  are  closed, 
its  stock  sells  "ex-dividend'' — that  is,  the  last  holder  gets  the 
dividend. 

Stock  is  sold  "cash"  for  deHvery  the  same  day;  "regular," 
to 'be  delivered  the  day  following;  "at  three  days";  or  at 
"buyer's  option"  or  "seller's  option" — usually  4  to  60  days. 
Most  sales  are  "regular." 

Practically  all  speculative  exchange  business  is  on  "  margin." 
Thus,  on  January  21,  C.  Brown  may  order  his  broker  to  buy 
100  shares  D.  &  M.,  whenever  securable  at  186.  He  deposits 
10%  of  the  money,  or  $1860  (called  the  margin).  The 
broker  supplies  the  rest  of  the  money  needed.  The  stock  is 
bought  the  same  day.  On  January  29,  when  the  stock  is  quoted 
at  189,  he  orders  the  broker  to  sell.  The  broker  charges  1/8% 
brokerage  each  way,  and  6%  interest  on  the  money  lent  to 
complete  the  transaction,  rendering  a  statement  as  follows: 


344 


BUSINESS  ARITHMETIC. 


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STOCKS  AND  BONDS.  345 

EXERCISE. 

1.  On  March  17,  I  purchased  through  a  broker  600  sh.  Reading,  at 
163 1 — buyer's  option  at  twenty  days.  I  called  for  the  stock  March  26, 
and  directed  ita  sale  at  164j.     Brokerage  1/8.     Determine  my  net  profit. 

2.  On  May  9,  my  broker  bought  for  me  800  sh.  P.  R.  R.  at  123|  on 
10%  margin,  selling  it  May  14  at  12o|.     Interest,  6%;  profit,  $  ?  . 

3.  On  June  9  a  broker  sold  short,  on  10%  margin,  500  sh.  N.  P.  at 
125  f.  The  stock  declined  so  that  on  June  16,  his  principal  ordered  his 
sale  covered  (121 1).     The  principal's  net  gain  was  $  ?. 

4.  Prepare  a  broker's  statement  for  his  principal,  A.  C.  Benson,  from 
the  following  memoranda:  Margin  10%;  interest,  6%.  Bought:  Jan.  11, 
200  sh.  N.  Y.  C,  regular,  106|;  Jan.  18,  500  sh.  N.  P.,  regular,  127 1; 
Sold:  Jan.  22,  500  sh.  N.  P.,  131f ;  Jan.  23,  200  sh.  N.  Y.  C,  105^. 

Collateral  Loans.  Many  business  men  invest  surplus  funds 
in  dividend-paying  stocks  and  bonds.  In  times  of  financial 
stringency,  they  borrow  of  their  banks  on  time  or  demand, 
depositing  this  stock  as  security  or  collateral.  These  stocks 
and  bonds  are  accepted,  if  reliable,  on  a  margin  of  10%  to 
20%  (i.  e.,  at  10%  or  20%  below  market  quotation).  No 
margin  is  required  on  government  bonds.  In  case  of  failure 
to  pay  the  loan,  the  securities  are  sold  by  the  bank,  the  debt 
is  cancelled  and  any  balance  is  returned  to  the  borrower. 
If  the  market  value  of  the  securities  falls  while  they  are  being 
held  as  collateral,  additional  collateral  must  be  deposited  or  a 
portion  of  the  loan  repaid. 

Stock  brokers  use  the  certificates  of  stocks  and  the  bonds 
they  buy  for  customers  as  temporary  collateral  for  loans  of 
this  order — termed  call  loans.  The  banks  lend  at  the  pre- 
vailing money  rate  and  have  the  privilege  of  changing  the 
rate  from  day  to  day  as  the  financial  market  varies.  They 
may  also  "call''  for  payment  at  any  time.  Rates  are  "easy" 
at  1%  to  3%;  "firm"  at  6%  to  8%,  and  "stringent"  at  higher 
rates.  During  a  marked  panic  some  money  was  lent  at  the 
rate  of  100%. 


346 


BUSINESS  ARITHMETIC. 


Face  of  a  Collateral  Note. 

$ Washington,  D.  C 190 . . . 

promise  to  pay  to 

The  National  Bank  of  Washington  City,  or  order 

at  said  Bank Dollars 

for  value  received  with  interest  at per  centum  per  annum,  having 

deposited  with  said  Bank,  as  collateral  security  for  the  payment  of  the 
full  sum  of  principal,  with  interest  and  cost  due  on  this  note,  and  also  as 
collateral  security  for  all  other  present  or  future  demands,  of  any  and  all 
kinds,  of  the  said  Bank,  against  the  undersigned,  due  or  not  due,  the 
following  to- wit : 

with  full  power  and  authority  to  said  bank  to  sell  the  whole  or  any  part  of 
said  security,  or  any  substitutes  therefor,  or  any  additions'  thereto,  at 
public  or  private  sale,  at  any  time,  and  at  the  option  of  said  Bank  or  its 
assigns,  on  the  non-performance  of  this  promise  or  any  part  thereof,  or 
the  non-payment  of  any  other  present  or  future  demands  of  said  bank  as 
aforesaid,  and  without  advertisement  or  notice  to  the  undersigned;  and 
upon  such  sale  the  holder  thereof  may  purchase  all  or  any  part  of  said 
securities,  discharged  from  any  right  of  redemption.  After  deducting  all 
proper  costs  and  expenses,  the  residue  of  the  proceeds  of  sale  shall  be  applied 
to  the  payment  of  the  note,  and  of  any  other  present  or  future  demands  of 
said  Bank  as  aforesaid,  and  the  undersigned  agrees  to  remain  liable  for  any 
deficiency  then  remaining.  In  case  of  depreciation  in  market  value  of  said 
security  at  any  time  pledged  for  this  loan  a  payment  shall  be  made  on 
account,  or  additional  security  added,  as  required  by  said  Bank. 


WRITTEN    EXERCISE. 
1.     Does  this  record  of  a  call  loan  show  sufficient  security,  if  the  margin 
must  be  10%? 


Loan  No.  86739. 
FoHo  No.  472. 
Demand  loan  for  $300,000. 
To  Robert  Harlenan. 


The     National  Bank. 

New  York,  February  17,  190— 


400 
1000 


200 
400 
500 
200 
800 
100 


111.  Central 
B.  &0. 
D.  L.  &  W. 
Pennsylvania 
Reading 
S.  Pac. 


127 
106 

388 
127 
167 
108 


STOCKS  AND  BONDS.  347 

Note.  In  reckoning  value,  fractions  of  quotations  are  omitted,  and 
final  value  written  to  thousands  and  hundreds  of  dollars.  The  first  item 
of  value  is  $25,400. 

Alterations  of  collateral  are  shown  by  quantities  in  italics. 

2.  Could  any  line  of  above  stock  be  withdrawn  without  destroying 
margin? 

3.  In  lots  of  100  sh.,  how  much  N.  Y.  C,  at  106,  might  be  substituted 
for  the  D.  L.  &  W.? 

4.  Compute  the  interest  on  this  demand  loan  at  3%,  Feb.  17-24; 
3|%,  Feb.  25-26;  4%,  to  Mar.  3  when  ''called." 

5.  On  Mar.  11,  $70,000  is  borrowed  at  3%,  on  400  sh.  N.  Y.  N.  H.  &  H., 
at  202,  as  collateral.  During  a  violent  break  in  the  market  stock  falls 
to  154.  How  much  additional  collateral  should  be  demanded  in  order  to 
preserve  a  10%  margin? 

6.  A  call  loan  of  $200,000  is  secured  by  collateral  of  $80,000  U.  S. 
bonds  (103);  100  sh.  111.  C.  (127);  200  sh.  Penn.  (128);  500  sh.  U.  P.  (160), 
200  sh.  N.  Y.  C.  (106)  and  300  sh.  C.  P.  (265).  If  the  borrower  pays 
$50,000  of  the  loan,  what  collateral  may  be  released?  Allow  10%  margin 
except  on  Government  bonds. 

Incomes  and  Investments.  The  investor  in  stocks  and 
bonds,  as  distinct  from  the  speculator,  is  looking  for  a  steady 
and  safe  return  from  his  money.  He  must  be  familiar  with 
factors  that  affect  quotations,  but  he  does  not  look  at  them 
in  the  same  light.  As  a  rule,  he  desires  a  fair  interest  on  his 
money,  security  against  loss,  and  a  market  for  disposal  in 
case  of  necessity.  In  all  cases,  it  is  well  to  have  the  advice  of 
a  reliable  banker  or  broker. 

Government  securities,  in  the  form  of  national  or  state 
bonds,  are  usually  safe  investments  and  are  easily  marketable, 
but  they  yield  relatively  low  returns.  Many  municipal  and 
county  bonds  are  also  reliable,  but  before  buying  it  is  wise  to 
see  what  security  is  back  of  them  and  how  regular  has  been 
the  payment  of  past  interest.  Railway  and  industrial  bonds 
should  be  examined  even  more  strictly  along  the  same  lines. 

In  buying,  one  should  note  the  date  of  maturity  of  the  issue, 
especially  if  he  desires  a  long  term  investment.     If  the  quo- 


348  BUSINESS  ARITHMETIC. 

tation  is  at  a  premium,  he  should  remember  that  the  premium 
represents  a  portion  of  his  investment  which  will  not  be 
repaid  him  when,  at  maturity,  the  bonds  are  settled.  The 
nearer  the  maturity  of  the  bonds,  the  greater  is  the  effect  of 
this  premium. 

In  buying  stocks,  one  should  invest  either  in  local  organi- 
zations, whose  policy,  standing  and  resources  he  can  examine, 
or  in  organizations  of  national  reputation,  whose  stock  has 
shown  steadiness  through  a  long  period,  and  which  is  readily 
marketable.  Before  buying,  one  should  look  up  quotations 
for  months  past  to  be  sure  that  the  current  quotation  is  not 
far  above  the  ruUng  price  because  of  some  temporary  market 
influence.  Times  of  depression  are  excellent  periods  for  the 
purchase  of  sound  stocks.  One  should  be  careful  not  to  be 
misled  by  high  rates  of  dividends.  The  safest  investments  on 
the  market  seldom  pay  over  5%  or  6%.  It  is  well  to  investi- 
gate and  determine  the  average  dividend  for  years  past,  to 
discover  whether  the  rate  is  steady,  steadily  increasing,  or 
decreasing,  etc. 

ORAL    EXERCISE. 

1.  I  can  buy  one  stock,  paying  6%  dividen'Is,  at  120,  and  another, 
paying  the  same  rate,  for  150.  How  many  dollars  per  share  income  in 
each  case?  Based  on  these  rates,  which  is  the  better  investment?  Suggest 
a  method  of  comparing  the  value  of  stock  investments. 

2.  I  have  $20,000  to  invest.  A  good  8%  stock  can  be  purchased  at 
200  and  a  5%  stock  at  80.  The  money  will  buy  how  many  shares  of  either 
kind?  What  is  the  income  in  each  case?  How  may  one  compare  invest- 
ments, knowing  the  sum  to  be  invested? 

3.  Investments  in  bonds  are  to  be  made  to  secure  a  yearly  income  of 
$720.  What  par  value  of  6%  bonds  will  produce  it?  Of  8%  bonds? 
The  6%  bonds  can  be  bought  at  par;  the  8%  bonds  at  120.  Security 
being  equal,  which  is  the  better  investment?  How  is  the  comparison 
made  in  this  case? 

Illustration.  Compare  the  value  of  C.  B.&  M.  4s  at  101  f,  and 
P.  R.  R.  4J-S  at  1071,  for  an  investment  of  $20,000. 


STOCKS  AND  BONDS.  349 

Solviion. 

C.  B.  &  M.  costs  101  f  +  1/8,  or  lOU. 

A  $1000  bond  will  cost  10  X  $101^,  or  $1015 

P.  R.  R.  costs  1071  +  i,  or  $108. 

A  $1000  bond  costs  10  X  $108,  or  $1080. 

$20,000  ^  $1015  =  19  and  $715  remaining. 

Therefore  19  C.  B.  &  M.  bonds  could  be  purchased. 

$20,000  ^  $1080  =  18  and  $560  remaming. 

Therefore  18  P.  R.  R.  bonds  could  be  purchased. 

19  X  $40  =  $760,  income  from  C.  B.  &  M.  4s. 

18  X  $45  =  $810,  income  from  P.  R.  R.  4|s. 

$4  -^  $101.5  =  .0394  (3.94%),  rate  income  on  C.  B.  &  M.  bonds. 

$4.5  ^  108  =  .0416  (4.16%),  rate  of  income  on  P.  R.  R.  bonds. 
The  P.  R.  R.  bonds  are  evidently  the  better  investment  if  long  term. 

EXERCISE. 

1.  What  income  will  be  received  from  the  investment  of  so  much  as  is 
possible  of  $60,000  in  U.  S.  4s  at  104^,  brok.  i? 

2.  Erie  4s,  at  108,  brok.  1/8,  earn  ?  %  on  investment. 

3.  Compare  the  investment  values  of  L.  &  N.  stock  at  145,  yielding 
6%,  and  Pullman  Palace  Car  Co.  stock,  at  181,  earning  8%. 

4.  Brown  has  $60,000  to  invest.  His  choice  narrows  down  to  T.  R.  R. 
at  141  (7%);  B.  &  O.  4%  pfd.  at  80,  and  D.  L.  &  W.  (10%,)  at  485.  Com- 
pare investment  values. 

5.  In  the  latter  part  of  1907  a  serious  financial  stringency  caused  heavy 
breaks  in  stock  quotations.  This  break  was  finally  stayed  on  Oct.  26. 
From  the  following  quotations,  find  the  rates  of  income  on  investments 
based  on  each  of  the  quotations  (high,  low  and  Oct.  26).  Show  that 
times  of  depression  are  good  periods  for  investment  in  sound  stocks. 

1907. 

High.  Low.  Stock.  Oct.  26.       Reg.  Dividend. 

145i  92f  L.  &  N 95  6%. 

75i  40  Mackay  Co 48  4 

14G  1001  Manhattan  Elev 103^  7 

92|  48  Mo.  Pac 51|  5 

147  108  Nash.  &  Chat 115  6 

117f  103  Nat.  Bis.  pfd 103  7 

1341  961  N.  Y.  C 991  6 

921  56  Nor.  &  Western 61|  5 

141-1  1131  Penn.  R.  R 115  7 

98|  72^  People's  Gas  &  Coal 73f  6 


350  BUSINESS  ARITHMETIC. 

Convertible  Bonds.  Some  railroad  corporations,  and  a  few 
other  organizations,  have  issued,  of  late,  series  of  bonds  that 
are  convertible,  at  a  fixed  ratio,  into  shares  of  stock.  Thus  a 
certain  company  issues  4%  bonds  that  may  be  exchanged  for 
stock  at  a  quotation  of  150  (par  100).  The  bonds  offer  a  safe 
but  low  rate  of  income.  If  the  business  gains  in  prosperity 
the  dividends  on  the  stock  may  increase  to  such  a  point  that  a 
greater  rate  of  income,  though  less  secure,  may  be  obtained 
by  conversion.  Questions  of  control,  speculation,  etc.,  may 
lead,  also,  to  conversion.  The  market  price  of  both  stocks 
and  bonds  also  has  its  effect. 

Illustration.  Mansfield  R.  R.  5%  convertible  bonds  are  exchangeable 
for  stock  at  120.  The  bonds  were  bought  at  108,  the  stock  is  quoted  at 
136  and  yields  8%  dividends.  Study  the  efifect  of  converting  $6000  par 
of  the  bonds. 

Solution. 

$6000  -v-  120  =  50,  number  of  shares  by  conversion. 
5%  of  $6000  =  $300,  income  from  bonds. 
8%  of  $5000  =  $400,  income  from  stock. 

Increase  of  income  by  conversion,  $100,  or  33^%. 

60  X  $108  =  $6480,  original  cost  of  bond  investment. 

50  X  $136  =  $6800,  present  selling  value  of  equivalent  stock. 

Direct  gain  by  converting  and  selling,  $6800  -  $6480  =  $320. 

Note.  This  does  not  take  into  account  the  effect  of  the  time  bonds 
have  to  run  before  maturing.  Bankers  have  "bond  tables"  to  show  the 
real  investment  value  of  bonds. 

EXERCISE. 

1.  $12,000  (par)  of  5%  bonds  may  be  converted  into  ?  shares  of  stock, 
if  exchange  quotation  of  stock  is  150. 

2.  $30,000  of  4%  bonds,  bought  at  92,  are  converted  into  stock  at  a 
quotation  of  120.  The  stock  is  sold  at  the  market  quotation  of  146 J. 
Compute  net  gain  or  loss. 

3.  M.  T.  R.  R.  Conv.  Gold  3^'s,  due  1925,  are  convertible  into 
stock  at  75  (par  50).  Bought  at  90,  the  bonds  yield  ?  %  on  investment. 
Compare  market  value  by  conversion,  if  stock  quotation  is  122f. 


STOCKS  AND  BONDS.  351 

4.  N.  P.  Conv.  4s,  of  1927,  at  86,  yield  ?  %  on  investment.  $80,000 
(par)  of  bonds  are  convertible  into  ?  shares  at  175.  Compare  investment 
values  if  9%  dividend  is  earned  by  stock. 

5.  R.  M.  Tr.  Conv.  5s  at  98  are  convertible  at  par.  The  last  stock 
quotation  is  85 f  and  the  last  dividend  4%.  Discuss  advisability  of  ex- 
change. 

Building  Loan  Associations.  Building  loan  associations 
are  a  form  of  bank  organized  under  state  laws  to  encourage 
the  accumulation  of  money  among  people  of  limited  means, 
with  the  object,  primarily,  of  their  making  real  estate  invest- 
ments and  building  homes.  They  originated  in  England  in 
1789.  The  first  association  of  the  kind  in  the  United  States 
was  organized  in  1831,  at  Frankford,  Pa.  Owing  to  many 
abuses  in  the  past,  laws  are  now  in  force  in  many  states, 
restricting  the  methods  and  operations  of  associations.  One 
state,  for  example,  limits  operating  expenses  to  2|%  of  in- 
debtedness. It  is  well  to  study  the  laws  of  the  incorporating 
state,  before  entering  any  association. 

The  capital  of  loan  associations,  consisting  of  the  accumu- 
lated savings  of  members,  is  divided  into  shares  of  $50  to  $250, 
usually  $200.  Each  share  runs  until  the  sum  of  installments 
paid  and  dividends  earned  equals  its  maturing  value.  The 
shares  are  issued  in  series,  at  periodic  intervals,  say  six  months 
or  a  year.  A  member  may  buy  any  number  of  shares  up  to 
a  fixed  limit  paying  by  small  weekly  installments.  He  receives 
the  face  value  of  the  shares  at  maturity. 

In  most  organizations,  borrowers  must  be  shareholders. 
They  may  borrow,  on  their  shares  as  security,  up  to  90%  of  the 
sum  paid  in.  The  intending  builder,  who  borrows,  usually  has 
a  piece  of  land,  which  is  pledged  with  hi§  shares  for  his  building 
loan.  Here  the  maturing  value  of  his  shares  is  commonly 
taken.  Usually  a  shareholder  may  withdraw  his  money  at 
any  time,  although  he  is  penalized  by  having  his  share  of 
earnings  reduced. 

Often  there  are  demands  from  members  for  more  funds  than 


352  BUSINESS  ARITHMETIC. 

are  available  for  lending.  It  has  become  common,  therefore, 
to  auction  off  the  money  in  the  treasury.  There  may  be  a 
fixed  minimum  of  interest,  say  5%,  and  the  one  who  bids  the 
highest  rate  over  5%  gets  the  money.  In  other  cases  the 
rate  is  fixed,  the  highest  bidder  being  the  one  who  offers  the 
largest  number  of  months  interest  in  advance.  This  advance 
interest  is  subtracted  from  the  face  of  the  loan  when  the  money 
is  paid  over.  Rates  are  often  higher  than  5%,  but  since  the 
stockholder  has  a  share  in  the  profits,  the  net  rate  is  reason- 
able. 

The  repayment  of  loans,  as  in  the  case  of  other  features, 
varies  in  different  associations.  Often  the  payments  are 
monthly,  and  for  equal  amounts,  but  the  proportion  going  to 
reduce  the  principal  steadily  increases  and  the  interest  de- 
creases.    Fully  paid  shares  are  accepted  as  cancelling  the  loan. 

EXERCISE. 

1.  Recently,  there  were  2,029,927  persons  said  to  be  sharing  in  the 
benefits  of  5737  associations  in  the  United  States,  and  the  total  assets 
were  $618,795,414.  Find  the  average  membership  and  assets  per  associ- 
ation. 

2.  The  paid  in  value  of  Arnold's  ten  shares  is  $746.80.  He  borrows 
90%  of  this  value  on  his  personal  note,  receiving  $ . 

3.  Norton  subscribes  for  fifteen  one-hundred-dollar  shares  in  an 
association  where  the  rate  is  25c  per  week  per  share.  What  is  his  weekly 
payment?  Ignoring  dividends,  how  long  will  it  take  him  to  pay  for  his 
stock?     To  what  sum  is  he  entitled  at  maturity? 

4.  Norton  borrows,  on  real  estate  security,  the  face  of  his  shares,  at 
6|%.     What  is  the  quarterly  interest  on  his  loan? 

5.  Mason  borrows  $1,000,  to  be  repaid  at  the  rate  of  $10  per  week, 
for  138  weeks,  and  a  final  jfeyment  of  $7.79.  What  sum  does  he  repay  in 
all?    What  is  the  average  rate  of  simple  interest  on  his  loan? 

6.  Randolph  borrows  $2,000  at  6%,  on  a  bid  of  32  weeks'  advance 
interest.     What  proceeds  of  his  loan  are  paid  him? 

7.  Anderson  wishes  to  obtain  $4,000.  He  applies  for  twenty  shares 
(par,  $200),  agreeing  to  pay  the  usual  dues  of  $1.00  per  month  per  share, 


STOCKS  AND  BONDS.  ^3 

and  1^%  of  dues  additional,  as  premium.     If  the  rate  of  loan  is  6%,  what 
is  his  total  monthly  payment? 

8.  An  association  issues  one  thousand  $200  shares  on  payments  of 
$1.00  per  month.  What  sum  has  it  to  lend  the  first  month?  Including 
8%  interest  on  the  first  month's  loans,  what  has  it  to  lend  the  second 
month?  Construct  a  table  showing  compounded  funds  to  the  end  of  the 
sixth  month,  and  the  book  value  and  profits  of  each  share  in  the  company. 

Distribution  of  Profits.  When  only  one  series  of  shares  is 
issued,  the  distribution  of  profits  for  any  one  year  consists 
simply  in  dividing  the  profits  by  the  number  of  shares  to 
determine  the  profits  per  share.  If,  however,  one  series 
follows  another,  at  short  intervals,  the  problem  is  complicated. 
Many  methods  are  followed,  the  most  common  being  that  of 
averaging. 

Illustration.  An  association,  at  the  end  of  its  third  year,  has  in 
force  three  annual  series  of  1,000,  500  and  1,000  shares  respectively.  The 
payments  are  $1.00  per  month  per  share.  Its  net  profits  to  the  time  of 
issue  of  the  fourth  series  are  $4860.     What  profits  are  due  each  share? 

Solution. 

The  first  dollar  on  a  share,  in  series  (1)  has  run  36  months,  the  second 
35  months,  and  the  last  1  month,  an  average  time  of  18^  months.  In  a 
similar  way,  the  average  dollar  in  series  (2)  has  run  12^  months,  and  in 
series  (3),  6^  months.  $36,  $24,  and  $12  have  been  paid  in  respectively, 
on  shares  of  the  three  issues. 

Series  (1)  1000  shares,  @  $36  yield  $36,000,  on  an 

average  of  18^  mo.  =  18|  X  36,000  =  $666,000  for  1  mo. 

(2)  500  shares,  @  24  yield    72,000,  on  an 

average  of  12|  mo.  12|  X  24,000  =  $150,000  for  1  mo. 

(3)  1000  shares,  @  12  yield  $12,000,  on  an 

average  of  6^  mo.  =  6^  X  12,000  =  $  78,000  for  1  mo. 

Equivalent  for  1  mo.  $894,000. 

a    •      /IN       -A     ^^      u         666,000.  „,  666  , 

Senes  (1)  evidently  shares  m  profits,  or  gqT-QQQ  per  share. 

/ON      -A     .1      V,         150,000  .  ^^  300  , 

(2)  evidently  shares  gg^QQQ  m  profits,  or  gg^QQQ  per  share. 

78  000  78 

(3)  evidently  shares  gQ^QQ^in  profits,  or  gg^Q^Q  per  share. 

^        of  $4860  =  $.0054. 


894,000 


354  BUSINESS  ARITHMETIC. 

Profits  per  share: 

Series  (1)  666  X  $.0054,  or  S3. 596-4. 
Series  (2)  300  X  .0054,  or  1.6200. 
Series  (3),    78  X  .0054,  or      .4212. 

EXERCISE. 

1.  At  jhe  end  of  one  year,  the  book  value  of  one  series  of  an  association 
is  $12.39,  and  of  a  second  series,  $6.10.  Series  (1)  has  2,000  shares  and 
series  (2)  1200  shares.  The  year's  profits  are  $1860.  Apportion  these 
profits  according  to  book  values. 

2.  The  Martinsburg  Building  and  Loan  Asso.  has  issued  five  annual 
series  as  follows:  (1)  2,000  shares;  (2)  5,000  shares;  (3)  3500  shares;  (4) 
1600  shares;  (5),  just  out,  1800  shares.  Dues  $1.00  per  month.  Ap- 
portion profits  to  date  of  $11,456. 


CHAPTER   XL. 

DOMESTIC  AND  FOREIGN  EXCHANGE. 

Domestic  Exchange. 

INTRODUCTORY    EXERCISE. 

1.  Name  common  reasons  for  sending  money  to  persons  in  other 
localities. 

2.  In  what  different  ways  may  a  person  send  the  actual  money? 

3.  Name  objections  to  sending  the  actual  money. 

4.  To  what  extent  are  the  postal  authorities  hable  for  money  sent  by 
registered  mail? 

5.  Who  is  liable  for  losses  of  money  sent  by  express  or  messenger? 

6.  In  what  ways,  to  your  knowledge,  may  payment  be  made  without 
sending  the  actual  mone}^?  Descnbe  such  as  you  are  familiar  with,  and 
state  advantages  and  disadvantages  of  each. 

7.  Is  it  cheaper  to  send  the  actual  money?     Is  it  safer? 

The  process  of  making  payments,  or  of  settling  accounts 
due  at  other  localities,  without  sending  the  actual  money,  is 
termed  exchange.  Exchange  within  a  country  is  termed 
domestic  or  inland  exchange. 

The  credit  papers  commonly  used  in  effecting  exchange  are 
money  orders,  checks  and  drafts. 

Money  Orders.  Money  orders  are  issued  by  the  National 
Government  and  by  express  companies.  They  are  really 
drafts  or  orders,  issued  at  one  office,  drawn  on  the  office  in 
the  locality  in  which  it  is  desired  to  make  payment,  and  pay- 
able, as  a  rule,  on  presentation.  Identification  before  payment 
may  be  required. 

The  funds  paid  by  the  purchaser  of  the  money  order  are  not 
transferred  by  government  or  express  company  to  the  office 
of  payment,  but  payment  at  that  office  is  made  from  any 

355 


356 


BUSINESS  ARITHMETIC. 


funds  on  hand.  Proper  bookkeeping  records  of  the  transfer 
are  kept.  Both  express  and  postal  orders  may  be  deposited 
Hke  checks  for  collection.  Many  hotels  accept  and  cash  them. 
They  may  be  transferred  by  endorsement — a  single  endorse- 
ment being  allowed  on  postal  orders,  while  express  orders  are 
freely  negotiable. 

If  more  than  one  hundred  dollars  is  to  be  sent  by  postal 
order,  additional  orders  must  be  taken  out.  The  individual 
express  money  order  is  limited  to  fifty  dollars. 

Express  Money  Order. 


6-0000000 


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dVAOCNT  AT  POINT  OF  l&SUt 

Pay  to  twf  nanrR  nFO'T'^^::^'tS-:?^A,Svf>^'"^^ ^£ 


.   *&REE.S-IOtP»NSMIT  AND 


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6-0000000- 

AaCRICAN  EXPRESS  CO. 


Mm- 


Note.     The  cost  of  an  order  equals  the  face  plus  the  fee. 


Fees  for  Money  Orders  Drawn  on  Domestic  Form. 
Payable  in  the  United  States  and  its  possessions;  also  for  orders  payable 
in  Bermuda,  British  Guiana,  British  Honduras,  Canada,  Cuba,  Mexico, 
Newfoundland,  the  United  States  Postal  Agency  at  Shanghai  (China), 
the  Bahama  Islands,  and  certain  other  Islands  in  the  West  Indies. 
For  Orders  From  $  0.01  to  $    2.50     3  cents. 

From  %  2.51  to  $     5.00     5  cents. 

From  $  5.01  to  $  10.00     8  cents. 

From  $10.01  to  $  20.00     10  cents. 

From  $20.01  to  $  30.00     12  cents. 

From  $30.01  to  $  40.00     15  cents. 

From  $40.01  to  $  50.00     18  cents. 

From  $50.01  to  $  60.00     20  cents. 

From  $60.01  to  $  75.00     25  cents. 

From  $75.01  to  $100.00     30  cents. 


DOMESTIC  AND  FOREIGN  EXCHANGE.  357 

Telegraph  Orders. 
Telegraph  orders  are  obtainable  in  the  larger  cities.  The 
money  is  paid  in  at  the  sending  office,  and  a  telegram  is  sent  to 
the  telegraph  company's  agent  at  the  place  of  payment,  direct- 
ing him  to  pay  the  sum,  from  any  funds  on  hand,  to  the  person 
specified,  on  identification.  The  rates  are  usually  one  per 
cent,  of  the  face  of  the  order  plus  twice  the  cost  of  a  ten  word 
message.  The  message  is  frequently  sent  in  the  Company's 
private  cypher  or  code,  each  word  standing  for  a  phrase. 

EXERCISE. 

1.  Let  some  pupil  describe,  from  personal  experience,  the  purchase  of 
a  money  order,  giving  the  steps  to  the  process. 

2.  Let  some  pupil  describe,  from  personal  experience,  the  cashing  of 
a  money  order. 

3.  Compute  the  total  cost  of  money  orders  to  cover  each  of  the  following 
amounts.  Give  the  denominations  of  money  orders  where  more  than  one 
required  to  make  a  payment.  Amounts:  $1.64,  $5.00,  $7.00,  $L50,  $25.00, 
$48.00,  $65.50,  $200.00,  $165.00,  $78.50,  $2.50,  $37.28,  $96.84,  $299.00. 

4.  An  express  agent  issues  money  orders  on  a  certain  day  for  $34.16, 
$345.,  $23.69,  $50.,  $76.15,  $20.25  and  $107.10.  He  started  the  day  with 
a  balance  in  his  money  order  drawer  of  $124.00.  During  the  day  he  cashed 
orders  for  $23.90,  $45.50,  $120.00,  $13.00,  $112.00,  $3.26  and  $1.56.  What 
should  be  his  balance  in  the  drawer  at  the  close  of  the  day? 

5.  Determine  the  individual  and  total  cost  of  these  telegraph  orders 
from  Washington,  the  rate  being  1%:  (a)  To  Philadelphia  $600.00,  plus 
message  cost  of  50  cents;  (6)  To  New  York,  $850,  message  50  cents;  (c) 
To  Newark,  Del.,  $386.50,  message  40  cents;  (d)  To  Indianapolis,  $590; 
message  80  cents  and  to  New  Orleans,  $2500,  message  75  cents. 

Checks.  It  is  customary  for  business  men  and  others  to 
keep  the  major  portion  of  their  cash  funds  on  deposit  in  banks 
or  trust  companies.  They  then  make  practically  all  payments 
by  checks,  or  orders  on  the  bank  to  pay  to  the  person  named  in 
the  check  the  sum  specified. 


358  BUSINESS  ARITHMETIC. 

Form  op  Check. 

GIRARD    XRUSTT    CONIRANY 


Note.  In  this  instance,  John  Doe  desires  to  make  a  payment  of  $400 
to  Henry  Newton.  Doe  draws  a  check  on  the  bank,  in  which  he  has 
money  deposited,  ordering  it  to  make  the  payment  to  Newton.  Newton 
may  either  take  the  check  to  the  bank  and  cash  it,  after  being  identified, 
or  he  may  deposit  it  with  his  own  bank,  after  endorsing  it,  and  the  bank 
will  collect  payment  from  the  drawer's  bank.  The  latter  bank  will  charge 
the  amount  against  Doe's  deposit. 

The  checks  of  others,  received  in  the  course  of  business  are 
deposited  to  the  credit  of  the  depositor's  account,  as  in  the 
case  of  actual  cash.  It  is  estimated  that  ninety  per  cent,  of 
all  business  transactions  involving  money  settlements  are 
settled  by  checks. 

In  commercial  banks,  checks  may  be  drawn  against  a 
deposit  at  any  time,  but  they  must  not  exceed,  at  any  time, 
the  depositor's  balance.     Checks  are  drawn  payable  (1)  '^To 

,  or  order" — in  this  case,- being  negotiable;  (2)  "To 

Cash" — as  a  rule  payable  to  the  depositor  himself;  (3)  "To 
Bearer" — payable  to   anyone  presenting  the  check;  or   (4) 

"To ,"  payable  only  to  the  person  named. 

Depositors  desiring  to  draw  money  for  a  specific  use,  as  for 
example,  to  pay  wages,  may  draw  checks  to  the  order  of  the 
particular  expense,  thus  "  Pay  to  the  order  of  Pay  Roll." 
Such  checks  are  cashed  by  the  depositor,  or  after  endorsement, 
by  his  special  representative.  In  some  localities  this  form  of 
check  is  now  forbidden. 

As  in  the  case  of  notes  (pp.  307,  308)  checks  may  be  trans- 
ferred by  endorsement  to  others  before  being  finally  cashed  or 
deposited. 

Business  men  deposit  in  their  own  banks,  practically  in  the 


DOMESTIC  AND  FOREIGN  EXCHANGE.  359 

same  manner  as  cash,  the  checks  received  by  them  from  others 
and  drawn  on  local  or  out-of-town  banks.  The  banks  re- 
ceiving these  checks  on  deposit  must  collect  payment  from 
the  banks  on  which  they  are  drawn.  To  avoid  the  necessity 
of  sending  messengers  daily  to  all  other  local  banks  on  which 
they,  hold  checks,  to  make  collections,  banks  form  local 
clearing  houses  at  w^hich  representatives  of  each  bank  meet 
daily  at  a  specified  hour.  Here,  in  a  systematic  way,  all 
checks  drawn  against  a  given  bank,  and  brought  by  repre- 
sentatives of  other  banks  are  charged  against  it,  and  all 
checks  it  presents  against  other  banks  are  credited  to  it. 
Balances  are  then  struck.  In  each  case  where  the  debits 
exceed  the  credits,  the  bank  concerned  pays  the  debit  balance 
to  the  clearing  house  officials  before  a  certain  time.  Immedi- 
ately after  the  receipt  of  this  money  from  the  banks  that  are 
said  to  owe  the  clearing  house,  it  is  paid  out  to  the  banks 
that  have  a  credit  balance.  The  clearing  house  thus  begins 
and  ends  the  day  without  money.  By  means  of  this  insti- 
tution balances  of  millions  are  sometimes  settled  by  the 
actual  payment  of  only  a  few  thousand  dollars. 

Illustration.  Rosslyn  has  three  banks.  On  a  certain  day  their 
representatives  meet.  The  First  National  Bank  presents  $12,000  in 
checks  against  the  Second  National  Bank,  and  $18,756.42  against  the 
Third  National  Bank.  The  Second  National  Bank  presents  $19,540  in 
checks  against  the  First  National  Bank  and  $15,126.10  against  the  Third 
National  Bank.  The  Third  National  Bank  presents  $14,1.18  in  checks 
against  the  First  National  Bank  and  $11,121.70  against  the  Second  National 
Bank.  The  Clearing  House  credits  each  bank  with  the  checks  it  presents 
and  charges  it  with  the  checks  drawn  on  it  and  presented  by  other  banks, 
thus: 

Cr.  Bal. 

$11,544.40 


Bank. 

Charges. 

Credits. 

Dr.  Bal. 

First  Nat.  Bank 

$33,658.00 

$30,756.42 

$2,901.58 

Second  Nat.  Bank 

23,121.70 

34,666.10 

Third  Nat.  Bank 

33,882.52 

25,239.70 

8,642.82 

$11,544.40    $11,544.40 
The  Clearing  House  collects  the  amounts  of  the  debit  balances  against 
the  First  and  Third  National  and  pays  them  over  to  the  Second  National 
Bank. 


360  BUSINESS  ARITHMETIC. 

The  checks  that  each  bank  receives  from  the  clearing 
house  are  taken  back  by  its  messenger  and  charged  against 
the  accounts  of  the  persons  drawing  them. 

Each  bank  receives,  with  other  deposits,  many  checks 
drawn  on  out-of-town  banks.  It  may  collect  these  by  mail, 
direct  from  the  banks  on  which  they  are  drawn,  or,  as  is  the 
common  custom,  it  may  collect  them  through  its  correspondent 
bank  or  banks  in  other  localities,  commonly  in  the  larger  cities, 
with  which  it  keeps  accounts.  In  the  same  way,  banks  in  the 
villages  and  towns  of  its  neighborhood,  and  banks  of  other 
cities,  may  keep  accounts  with  it. 

Illustration.  Peter  Morton  draws  a  check  on  his  bank,  The  Idaho  Na- 
tional Bank  of  Washington,  in  favor  of  James  Rawson  of  Newark,  N.  J.,  and 
sends  it  to  the  latter  in  settlement  for  a  purchase  of  merchandise.  James 
Rawson  deposits  the  check  to  the  credit  of  his  account  in  his  home  bank, 
the  First  National.  The  First  National  Bank  credits  Rawson  and  sends 
the  check  to  its  correspondent  bank  nearest  Washington,  the  Fortieth 
National  Bank  of  Baltimore,  charging  that  bank.  The  Fortieth  National 
Bank  credits  the  Newark  bank,  and  sends  on  the  check,  properly  endorsed, 
to  its  correspondent  bank  in  Washington,  say  the  Twentieth  National 
Bank.  The  Twentieth  National  Bank  credits  the  Fortieth  National  Bank 
of  Baltimore,  and  exchanges  the  check  with  the  Idaho  National  Bank 
through  the  Clearing  House.  The  Idaho  National  Bank  charges  the 
check  against  Morton's  account. 

Thus  it  is  evident  that  all  the  checks  a  depositor  draws 
against  his  account,  and  gives  out  in  payment,  either  locally 
or  at  other  points,  find  their  way  back  to  his  own  bank  and 
are  charged  against  his  account.  Sometimes  they  take  very 
roundabout  paths  back,  but  the  whole  history  is  shown  by  the 
endorsements  on  the  back.  It  will  be  found  interesting  to 
study  the  endorsements  of  cancelled  checks. 

Of  course,  any  two  correspondent  banks  are  continually 
exchanging  checks  for  collection,  and  their  debit  or  credit 
balances  with  each  other  are  constantly  changing.  At  more 
or  less  periodic  intervals,  however,  these  balances  are  settled 
in  full  or  in  part  by  transfers  of  actual  currency. 


DOMESTIC  AND  FOREIGN  EXCHANGE.  361 

Collection  Fees.  Some  banks  charge  for  out-of-town  col- 
lections made  for  other  than  their  depositors,  and  sometimes 
for  collections  for  their  depositors.  The  fee  (rate  of  exchange) 
for  such  collections,  while  still  more  or  less  variable,  is  becoming 
steadily  more  uniform,  especially  in  the  East,  being  frequently 
only  1/10%.  Sometimes  there  is  a  minimum  charge  of  10 
cents  or  25  cents. 

Bank  Drafts  and  Cashiers'  Checks.  It  frequently  happens, 
owing  to  the  risk  of  loss,  that  objection  is  made  to  receiving 
the  check  of  a  private  depositor.  In  such  cases  the  payer 
may  have  his  check  certified  by  an  ofiicial  of  his  bank  on  which 
the  draft  is  drawn.  The  certification  is  stamped  across  the 
face  of  the  check  and  signed  by  the  cashier  or  other  bank 
oJBScer.  The  bank  thus  guarantees  the  check  and  protects 
itself  by  immediately  charging  the  check  against  the  depositor's 
account.  Such  checks  are  more  readily  accepted.  In  case, 
however,  a  certified  check  is  not  used,  it  should  not  be  de- 
stroyed, but  should  be  deposited  to  the  credit  of  the  account 
against  which  it  was  originally  drawn. 

The  payer  may  also  make  settlement  by  means  of  a  hank 
draft,  purchased  at  his  own  bank,  or  at  any  other  that  will 
accommodate  him,  and  drawn  on  some  correspondent  bank 
either  in  the  neighborhood  of  the  payee,  or  in  some  large 
central  city.  Exchange  on  New  York  or  Chicago  is  acceptable 
anywhere.  These  drafts  are  often  drawn  to  the  order  of  their 
purchaser  in  order  that  he  may  endorse  them  over  to  the 
intended  payee  and  thus  secure  proof  of  payment.  The  payee 
may  deposit  them  or  transfer  them  by  endorsement,  the  same 
as  checks.  The  charge  (rate  of  exchange)  is  usually  the 
face  plus  a  slight  premium,  generally  1/10%,  although  a  bank 
will  often  issue  drafts  to  its  regular  customers  at  a  cost  equal 
to  the  face. 

If  the  correspondent  or  paying  bank  has  funds  to  the  credit 
or  issuing  bank,  the  drafts  are  charged  against  these.     If  not, 


362  BUSINESS  ARITHMETIC. 

the  issuing  bank  will  send  bank  checks  to  its  correspondent 
for  collection,  or  will  give  the  bank  credit  and  balance  up  later. 
As  a  rule,  no  charge  is  made  for  the  collection  of  bank  drafts 
when  deposited  for  that  purpose,  if  they  are  drawn  on  banks 
at  the  large  money  centers. 

Form  of  Bank  Draft. 
No.  6129  Seatde,  Wash.,  Mar.  2,  1912. 

Johnson,  Morton  and  Co. 
Bankers 

Pay  to  the  order  of James  F.  Porter $826^*/^ 

Eight  hundred,  twenty-six  j%\ Dollars. 

To  Fifteenth  National  Bank,  Andrew  Norton, 

St.  Paul,  Minn.  Cashier. 

Form  of  Bank  Draft. 

In  place  of  a  bank  draft,  an  ordinary  cashier's  check  on  his 
own  bank  may  be  obtained  by  a  depositor,  its  cost  being 
charged  against  his  account.  Such  checks,  being  official  in 
character,  and  having  the  weight  of  a  bank  back  of  them, 
pass  more  readily  among  strangers  than  do  private  checks. 

Certificate  of  Deposit.  Instead  of  opening  a  regular  checking 
account,  one  may  deposit  money  in  a  bank  and  receive  therefor 
a  certificate  of  deposit,  transferable  by  endorsement.  This 
form  is  often  used  when  some  fixed  sum  of  money  is  to  be 
deposited  for  safe-keeping.  Some  savings  banks  and  trust 
companies  will  issue  certificates  for  fixed  periods,  such  as  one 
year,  bearing  interest  at  a  certain  rate,  usually  higher  than  that 
allowed  on  deposits  subject  to  check. 

A  Certificate  of  Deposit. 

$6000  T^iy  Chicago,  Illinois,  June  16,  191 — 

National  Bank 

This  certifies  that  Thomas  Medway  has  deposited 

Six  Thousand  j%% Dollars, 

payable  to  the  order  of C.  T.  Robinson 

on  the  return  of  this  certificate  properly  indorsed.  ,^      .    _    ^ 

Martin  Kentj 

Cashier. 


DOMESTIC  AND  FOREIGN  EXCHANGE.  363 

Commercial  Drafts.  Ordinary  commercial  drafts  are  fre- 
quently used  for  settlements  and  collections. 

$825tV7  Columbus,  Ohio,  Sept.  19,  191— 

At  sight Pay  to  the  order  of 

Ourselves 

.  . .  .Eight  hundred  twenty-five  ^^^ Dollars. . 

Value  received  and  charge  to  the  account  of 

To  Thompson  Bros.  Lewis  Mark  &  Son. 

No.  67    Baltimore,  Md. 

Note.  Thompson  Bros,  owe  Lewis  Mark  &  Son  $825.67.  When  the 
account  is  due  Lewis  Mark  &  Son  draw  a  sight  draft  on  Thompson  Bros, 
and  give  it  to  their  home  bank  to  collect.  The  Columbus  bank  sends 
the  draft  to  its  correspondent  in  Baltimore.  The  bank  in  Baltimore 
presents  the  draft  to  Thompson  Bros,  for  settlement.  When  payment  Is 
received  the  Columbus  bank  is  notified  and  it  credits  Lewis  Mark  &  Son's 
account  for  the  amount  less  a  small  fee  (collection  and  exchange). 

Time  or  sight  drafts  are  used  frequently  to  collect  for 
merchandise  at  time  of  delivery.  Goods  are  shipped  to  a 
customer  and  a  through  bill  of  lading  is  taken  from  the  trans- 
portation company.  This  is  a  receipt  for  the  goods  and  an 
agreement  to  deliver  them  to  the  consignee,  or  to  his  order.  The 
selling  firm  also  makes  out  a  sight  draft  on  the  customer,  if 
the  terms  are  cash  on  delivery,  or  a  time  draft,  if  credit  for  a 
certain  period  is  to  be  allowed.  The  draft  and  bill  of  lading 
are  fastened  together  and  given  to  the  seller's  bank.  The  bank 
endorses  the  draft  and  sends  it  to  its  correspondent  bank  in 
the  neighborhood  of  the  purchaser.  This  bank  presents  the 
draft  to  the  customer  for  payment,  if  a  sight  draft,  or  for 
acceptance  if  a  time  draft.  When  payment  or  acceptance  has 
been  made,  the  bank  surrenders  the  bill  of  lading  and  the 
customer  may  secure  his  goods.  Time  drafts  frequently  are 
left  in  the  bank  for  presentation  again  when  payment  becomes 
due. 

Rates  of  Exchange.  Postal  and  express  money  orders  cost 
more  than  their  face  value.  Bank  drafts  frequently  sell 
at,  or  above,  their  face  value.     Bank  and  commercial  drafts 


364  BUSINESS  ARITHMETIC. 

may  sell  for  less  than  their  face  value.  Exchange  is  at  par 
if  the  cost  of  a  draft  equals  its  face;  it  is  above  par,  or  at  a  pre- 
mium, if  the  cost  exceeds  the  face;  it  is  below  par,  or  at  a  dis- 
count, if  it  sells  for  less.  Exchange  on  small  sums  is  usually 
at  a  premium  to  repay  cost  of  handling.  The  rate  of  exchange 
varies  with  the  demand  for  money  at  certain  centers  of  trade 
and  with  the  cost  of  shipment  of  actual  currency. 

Illustration.  In  the  harvest  season  for  grain,  eastern  purchasers 
are  sending  checks  west  in  settlement  for  purchases.  Suppose,  for  example, 
that  large  orders  are  being  received  in  Chicago  from  New  York  customers. 
Checks  and  drafts  drawn  on  New  York  banks  and  presented  or  deposited 
in  Chicago  banks  will  tend  to  seriously  decrease,  or  to  more  than  exhaust, 
the  balances  to  the  credit  of  the  New  York  banks. ^  At  such  times  exchange 
on  Chicago  will  be  selling  at  a  premium  in  New  York,  because  the  New  York 
banks  selling  such  exchange  will  be  under  the  necessity  of  shipping  actual 
currency  to  Chicago  to  meet  the  drafts. 

At  the  same  time,  however,  Chicago  purchasers  will  find  exchange  on 
New  York  below  par,  because  Chicago  banks  have  so  much  due  them  from 
New  York,  that  drafts  on  the  latter  city  may  be  met  easily  without  the 
transfer  between  the  cities  of  the  actual  currency.  The  sale  of  such  drafts 
really  benefits  Chicago  by  lessening  the  balance  due  from  New  York  to 
Chicago. 

In  spring,  when  manufactured  and  imported  goods  are  going  out  from 
New  York  all  over  the  country,  and  merchants  all  over  the  country  owe 
New  York  dealers  and  manufacturers,  conditions  are  reversed.  There 
is  such  a  demand  for  exchange  on  New  York  that  the  rate  of  exchange  is 
almost  sure  to  go  above  par. 

The  rate  of  exchange  seldom  if  ever  goes  above  the  rate 
for  shipping  the  actual  money.  If  the  cost  of  shipping  money 
between  two  given  cities  is  $10  per  $10,000  the  rate  of  premium 
will  not  go  above  1/10%. 

Collection  rates  for  drafts  are  somewhat  arbitrary,  being  gov- 
erned by  general  trade  conditions.  As  a  rule,  however,  bank 
drafts  on  the  large  money  centers  are  collected  without  charge. 

EXERCISE. 
1.    Secure  a  bank  deposit  slip.     Find  out  how  to  make  a  deposit. 
Explain  the  process  of  opening  an  account  and  making  the  original  deposit. 


DOMESTIC  AND  FOREIGN  EXCHANGE. 


365 


2.  If  you  owed  a  man  in  Albany  a  payment  of  $760,  which  would 
probably  be  the  cheapest  safe  way  of  making  settlement? 

3.  C.  Y.  Portner  collects  through  his  bank,  at  a  charge  of  1/10%  for 
collection,  checks  on  New  York  for  $4.56.89  and  $887.50  and  on  Phila- 
delphia for  $1,456.     What  is  the  net  sum  to  be  credited  to  his  bank  account? 

4.  Extend  the  following  form  showing  collections  made  by  one  bank 
for  its  correspondent  bank.     Collection  1/10%. 

Fortieth  National  Bank. 

,  N.  Y.,  September  15,  191—. 

Mr.  Robert  T,  Brown,  Cashier. 

Washington,  D.  C,  Ninth  National  Bank, 
Dear  Sir: 

We  have  credited  your  account  this  day  for  the  proceeds  of  the  collections 
stated  below.  j^^^  Fadelet,  Cashier. 


Your  No. 

Payer. 

Amount             Collection.             Proceeds. 

456 

Robert  Chase 
A.  B.  Nortins 
Bruce  &  Co. 
Henry  Ames 
Besley  Bros. 

480 
1239 
2311 

477 
5465 

00 

80               

45               

62               

00               

- 

461 

511 

512 

589 

....           

.... 

5.  Banks  A,  B,  C  and  D  form  a  Clearing  House.  On  Sept.  16, 
Bank  A  presents  checks  against  Bank  B  for  $12,178.80;  against  Bank  C 
for  $4566.75;  against  Bank  D  for  $56,124.59.  Bank  B  presents  checks 
against  Bank  A  for  $45,000;  against  Bank  C  for  $34,123;  against  Bank  D  for 
$19,234.59.  Bank  C  presents  checks  against  Bank  A  for  $13,543;  against 
Bank  B  for  $24, 118.25 5  and  against  Bank  D  for  $7,188.55.  Bank  D  pre- 
sents checks  against  Bank  A  for  $12,325;  against  Bank  B  for  $35,455.60 
and  against  Bank  C  for  $41,157.20.  Find  the  debit  or  credit  balance  of 
each  bank  with  the  Clearing  House  and  determine  what  payments  must 
be  made  to  settle  all  balances. 

6.  What  is  the  net  sum  placed  to  the  credit  of  John  Grandfield  on 
deposits  of  checks  for  $435,  $350,  $527.50  and  $56.88,  on  which  a  collection 
fee  of  1/8%  is  charged? 

7.  Compute  the  cost  of  a  bank  draft  on  Richmond,  Va.,  for  $723.90, 
at  1/10%. 

8.  How  large  a  draft  may  be  purchased  for  $1127.13  at  1/10%  pre^ 
mium? 

9.  Determine  the  proceeds  from  the  collection  through  an  express 
company  of  an  account  of  $2,356,  at  a  rate  of  $1.75  per  $1,000. 


366  BUSINESS  ARITHMETIC. 

10.  Draw  up  a  check  on  an  assumed  bank  to  pay  for  a  bank  draft  of 
$2264  at  1/5%  premium. 

11.  Robert  Gaines  sold  H.  M.  Chilton  a  bill  of  goods  amounting  to 
$1764.80.  Gaines  drew  a  draft  in  settlement  and  sent  it,  with  the  bill  of 
lading  for  the  goods,  through  his  bank.  What  sum  was  added  to  the  credit 
of  his  account  when  the  bank  had  collected  the  draft,  its  charge  being  1/5%? 

12.  A.  C.  Crandall  sold  a  bill  of  goods  for  $7500  to  George  Grimes,  the 
terms  being  "30  day  acceptance."  •  Crandall  drew  the  draft  and  sent  it 
through  his  bank  for  acceptance.  After  acceptance,  the  bank  discounted 
the  draft  for  Crandall  at  1/10%  and  6%  interest.  What  proceeds  did 
Crandall  receive? 

13.  A  commercial  draft  for  $475  cost  $474.05.  Was  exchange  at  a 
premium  or  at  a  discount?     At  what  rate? 

14.  A.  M.  Beardsley  drew  a  draft  on  John  Evans,  for  $467.56  and 
collected  it  through  his  bank  at  a  rate  of  collection  of  1/20%.  What  sum 
should  be  credited  to  his  account? 

15.  Was  exchange  at  a  premium  or  at  a  discount,  and  at  what  rate,  if 
a  draft  for  $23,450  cost  $23,567.25? 

16.  Boston  is  selling  drafts  on  New  Orleans  at  3/4%  premium.  Which 
way  is  the  balance  of  trade?  What  is  the  cost  of  a  draft  on  New  Orleans 
face  value  of  $1245? 

17.  Suppose  the  balance  of  trade  between  Indianapolis  and  Pittsburg 
is  largely  in  favor  of  the  latter.  What  docs  this  mean?  In  which  city  is 
it  likely  that  exchange  will  be  at  a  premium?  Why?  At  a  discount? 
Why? 

18.  Name  several  cities  that  might  have  a  balance  of  trade  in  their 
favor  in  the  spring.    In  the  fall.     Give  your  reasons. 

FOREIGN  EXCHANGE. 

All  forms  of  payment  possible  under  domestic  exchange  are 
available  for  payments  abroad.  Theoretically,  foreign  ex- 
change is  based  on  the  same  general  principles  as  domestic 
exchange.  In  practice,  however,  all  computations  are  affected 
by  differences  in  currency  denominations  and  standards,  and 
by  methods  of  quoting  exchange.  From  the  banker's  stand- 
point, foreign  exchange  is,  in  many  ways,  the  most  intricate 
of  all  forms  of  banking. 


DOMESTIC  AND  FOREIGN  EXCHANGE.  367 

Rates  of  Exchange.  Foreign  rates  are  variously  expressed, 
but  may  be  broadly  classified  under  three  heads. 

1.  Rates  per  foreign  coin.  Thus  a  London  exchange  rate  of  "4.875" 
means  that  one  pound  sterling  can  be  bought  for  $4,875.  A  rate  of  "40f  " 
on  Amsterdam  means  that  one  guilder  exchange  on  that  city  costs  $.40 f. 

2.  Rates  on  a  set  number  of  coin.  Exchange  on  German  cities  is  usually- 
quoted  "per  four  marks."  Thus  a  quotation  of  "96,"  on  Berlin,  means 
that  a  four  marks  bill  of  exchange  costs  $.96. 

3.  Rates  in  "coin  per  dollar."  This  form  of  exchange  is  often  quoted 
for  France  and  other  Latin  countries,  as  Spain,  Italy,  Switzerland,  etc. 
Thus  a  rate  of  5.15 f  on  Paris  means  that  a  draft  on  Paris  for  5.15 f  francs 
may  be  purchased  for  one  dollar. 

Note.     Form   (1)   is  sometimes  used  for  exchange  on  any  country 
especially  in  connection  with  some  forms  of  money  orders. 
Illustrations. 

(1)  Compute  the  cost  of  a  £450  draft  on  London,  purchased  at  4.875. 
Solution. 

$4,875  Since    £1    cost    $4,875,    £450    will    cost 

450  450  X  $4,875. 

243750 
19500 


$2193.75,  Ans. 

(2)  Compute  the  cost  of  a  draft  on  Berlin  for  1200  M.,  purchased  at 

96. 
Solution. 

4)1200  In  1200  M.  there  are  300  sets  of  4  M. 

~~300,  no.  sets  4  M.         If  one  set  costs  96c,  300  sets  cost  300  X  96c. 
$0.96 
300 
$288.00  cost. 

(3)  Compute  the  cost  of  a  12,000  fr.  draft  on  Paris,  purchased  at  5.15. 
Solution. 

$  2330.09  Since  5.15  francs  cost  $1.00,  12,000  francs 

5.15)12000.  will  cost  as  many  dollars  as  5.15  is  contained 

1030  tmies  in  12,000. 

1700 
1545 
1550 
1545 
5000 
4635 


368 


BUSINESS  ARITHMETIC. 


Foreign  Money  Orders.  The  International  Postal  Money- 
Orders  and  the  express  orders,  which  are  similar  to  domestic 
money  orders,  are  issued  for  amounts  not  exceeding  one 
hundred  dollars,  and  are  usually  payable  in  the  money  of  the 
country  on  which  they  are  drawn.  Fees  for  this  service  are 
small. 

1.  What  will  an  express  money  order  for  £5  6s.  cost  at  $4.87? 

2.  How  large  a  draft  on  London  will  $50  buy,  at  the  same  quotation? 

3.  Robbins  wishes  to  send  a  friend  in  Rome  an  order  for  290.5  lira. 
At  19.5c  per  lira,  the  order  costs  $  ? 

4.  For  $40  one  may  buy  an  order  on  Berlin  for  ?  M.,  if  the  rate  is  24c 
per  M. 

Note.  Money  payments  may  be  made  by  cable  in  the  same  way  in 
which  they  are  made  by  telegraph.  In  cable  orders,  a  charge  is  made  for 
each  word  of  the  address. 

5.  At  20c  per  word,  and  1%  of  the  amount,  find  the  cost  of  a  24-word 
cable  order  on  London  for  £600,  at  a  rate  of  exchange  of  $4.8725. 

Travelers'  Checks.  Travelers  may  secure  from  express 
companies  and  banks  travelers'  checks,  which  can  be  cashed 

Form  of  Traveler's  Check. 


at  hotels  and  banks  abroad.  They  are  issued  for  fixed  amounts 
and  state  on  their  face  their  value  in  the  coinage  of  the  differ- 
ent countries  in  which  they  may  be  presented  for  "cashing." 


DOMESTIC  AND  FOREIGN  EXCHANGE.  369 

The  purchaser  signs  his  name  on  the  face  at  the  time  of 
purchase,  and  again  on  face  or  back,  for  purposes  of  identi- 
fication, at  time  of  cashing.  The  checks  are  sometimes  issued 
in  books  or  series. 

Letters  of  Credit.  A  traveler's  letter  of  credit  is  an 
instrument  issued  by  a  bank  to  its  representatives  abroad, 
authorizing  them  to  honor  drafts  of  the  person  named  in 
the  body  of  the  letter,  to  total  not  exceeding  an  amount 
stated. 

The  person  desiring  a  letter  of  credit  deposits  with  the 
issuing  bank  money  or  bonds  as  security.  On  receiving  the 
letter  of  credit,  he  places  his  .signature  on  the  face,  and  also 
gives  the  issuing  banker  other  specimens  of  his  signature, 
which  are  forwarded  to  the  correspondent  banks  with  the 
formal  notice  of  the  issuance  of  the  letter.  Once  abroad, 
whenever  the  holder  of  the  letter  desires  money,  he  stops  at 
any  correspondent  bank  named  in  the  letter  and  presents  his 
personal  draft  for  the  amount  desired.  After  the  signature 
has  been  verified,  he  is  paid  the  amount  in  the  coinage  of  the 
country.  This  amount  is  indorsed,  also,  on  the  letter  of  credit. 
When  the  "credit"  called  for  has  been  exhausted,  the  letter 
is  taken  up  by  the  bank  making  the  last  payment,  cancelled, 
and  returned  to  the  issuing  bank. 

EXERCISE. 

1.  Determine  the  cost  of  a  12,000  franc  letter  of  credit,  at  5.18  and  1% 
commission. 

2.  On  a  £600  letter  of  credit,  are  endorsed  payments  of  £120,  Nov.  10; 
£50,  Dec.  5;  £180,  Jan.  16;  £125,  Feb.  12.  Purchased  at  4.86  and  1% 
commission,  and  unused  portion  redeemed  at  same  rate,  what  repayment 
should  be  made  to  the  holder? 

Commercial  Letters  of  Credit.     These   letters    are    issued 
to  mercantile  houses  desiring:  to  import  goods  to  enable  them 
25 


370  BUSINESS  ARITHMETIC. 

to  get  credit  from  exporters  for  the  usual  periods  of  30,  60  or 
90  days.  On  proper  evidence  of  the  shipment  of  goods,  a 
foreign  bank  at  point  of  shipment,  will  buy  the  shipper's 
draft  against  the  person  named  in  the  letter,  at  the  prevailing 
rate  of  commercial  exchange,  reimbursing  itself  by  sending 
the  draft  to  party  drawn  on  for  collection,  or  by  selling  the  draft 
in  the  open  market.  Credits  are  issued  usually  in  pounds 
sterling  because  such  exchange  is  easily  negotiated  in  any 
part  of  the  world.  They  are  sometimes  issued  payable  in  the 
money  of  the  country  to  which  they  are  sent.  Commercial 
credits  are  issued  in  four  parts,  one  being  sent  to  the  exporter, 
one  to  the  bank  named  in  the  credit,  and  one  to  the  importer, 
or  purchaser,  the  fourth  being  retained.  Prices  vary  with 
credit  term,  financial  responsibility  of  applicant,  and  many 
other  conditions.  The  problems  of  computation  are  the  same 
as  those  involved  in  other  draft  transactions. 

Bills  of  Exchange.  Drafts  drawn  by  a  person  or  bank  in 
one  country  on  a  person  in  another  country  are  termed  bills 
of  exchange.  They  include  bankers',  commercial  and  docu- 
mentary bills.  Bankers'  bills  are  a  bank's  checks,  drawn 
against  its  deposits  in  foreign  banks.  Commercial  bills  are 
those  drawn  by  one  merchant  against  another.  Documentary 
bills  are  commercial  bills,  secured  by  bills  of  lading,  insurance 
policies,  or  other  documents  that  control  the  merchandise 
in  transit.  Bills  of  exchange  were  formerly  issued  in  triplicate. 
They  are  now  issued,  as  a  rule,  in  duplicate,  so  worded  that 
the  payment  of  either  one  makes  void  the  other.  Sometimes 
the  ordinary  single  draft  is  used.  Owing  to  delays  and  dangers 
of  transit,  the  duplicate  drafts  were  formerly  sent  by  different 
routes  or  mails.  Now  the  tendency  is  to  keep  the  duplicate 
on  file. 


2 


o  S 
ii  ^ 
a,  « 

I" 


DOMESTIC  AND  FOREIGN  EXCHANGE.  371 

A  Duplicate  Bill. 

PHiLADBLPmA,  Pa.,  April  19,  19 — 
2450  M. 

At  sight  of  this  second  of  Exchange  (first  unpaid),  pay  to 

the  order  of Ourselves 

Twenty-four  hundred  and  fifty  Marks, 

Value  received  and  charge  to  the  account  of 
To  K.  R.  Schmidt 

Berhn  S.  T.  Coles  &  Co. 

Simple  Foreign  Bill. 
£1000—  Philadelphia,  March  12th,  190— 

On  demand Pay  to  the  order  of 

John  J.  Doe 

One  thousand Pounds  sterling 

Value  received  and  charge  to  account  of 

To  J.  S.  Morgan  &  Co. 

No.  16    London,    England.  John  Smith. 


Fluctuation  of  Exchange.  The  rate  of  exchange  is  affected 
by  numerous  causes.  If  exported  goods  greatly  exceed  imports 
in  value,  the  rate  of  exchange  tends  to  lower  because  there  is 
less  demand  for  remittances  abroad.  If  more  money  is  due 
abroad  than  is  due  the  home  country,  the  rate  tends  to  rise. 
When  money  is  in  demand  here,  and  rates  rise,  rates  of 
exchange  may  fall  because  there  is  less  demand  for  bills.  If 
money  rates  are  high  abroad,  exchange  rates  may  rise  because 
of  the  demand  for  bills  in  order  to  send  money  to  a  place  of 
good  investment.  In  general,  the  question  of  supply  and 
demand  seriously  affects  the  quotations. 

The  discount  rates  on  commercial  bills,  at  foreign  money 
centers,  also  affect  the  rate  of  exchange.  If  rates  of  discount 
are  low,  commercial  drafts  may  "cash''  for  a  greater  sum  than 
could  be  realized  by  selling  them  as  exchange.  The  question 
of  the  parity  of  units  of  foreign  coinage,  as  compared  with 
the  units  of  our  own  coinage,  also  has  its  effect.  The  exchange 
rate  will  seldom  go  beyond  the  limits  of  the  cost  of  buying 


372  BUSINESS  ARITHMETIC. 

and  shipping  the  actual  gold — these  costs  including  commis- 
sion, freight,  insurance,  and  often  interest  lost  while  money 
is  in  transit.  That  is,  the  rate  of  exchange  does  not  go  beyond 
the  point  where  it  is  cheaper  to  send  the  actual  money. 

Newspaper  quotations  are  given,  usually,  for  both  sight 
and  time  paper.  Par  of  exchange  is  either  (1)  mint  par — the 
equality  of  actual  value  of  pure  metal  in  units  of  money  in  the 
countries  concerned ;  or  (2)  commercial  par — a  rarely  used  term 
referring  to  a  practical  equality  of  debt  between  countries  that 
reduces  the  rate  of  exchange  to  the  face  value  plus  the  cost 
of  shipping. 

ILLUSTRATIONS  OF  EXCHANGE. 

1.  ^,  in  New  York,  sells  B  in  London  merchandise,  $600.  A  draws  his 
draft  on  5,  in  pounds  sterling.  He  may  then  discount  the  draft  at  a  bank; 
or  he  may  sell  it  in  open  market  to  some  party  who  wishes  to  make  a  pay- 
ment in  England.  In  the  former  case,  the  bank  either  sends  the  draft  to 
England  for  collection — in  which  case  the  money  is  probably  held  by  the 
correspondent  bank  in  England,  and  the  home  bank  draws  and  sells  its 
bank  exchange  against  the  sum,  and  like  sums,  to  any  purchaser — or  it 
may  sell  the  original  draft  to  a  customer.  In  case  A  sells  the  original  draft 
direct,  or  through  a  broker,  the  party  buying  it,  C,  who  may  owe  D  in 
England,  sends  the  draft,  properly  endorsed,  to  the  latter,  who  presents 
it  to  B  for  payment.  By  such  a  general  process,  it  is  evident  that  A  gets 
his  payment  at  once,  and  that  sooner  or  later  B  pays  the  draft  on  pres- 
entation and  thus  settles  for  the  goods  that  he  bought.  Incidentally,  the  bill 
has  served  the  further  purpose  of  making  settlements  between  other  parties. 

2.  In  case  A  had  been  the  buyer  and  B  the  seller,  B  might  carry  through 
the  same  process  shown  above.  The  settlement  might  start  with  A, 
however,  who  might  go  to  a  bank  or  broker,  and  buy  a  draft  on  England 
for  the  amount  due,  sending  it  to  B  who  would  cash  it.  In  this  case  A 
corresponds  to  C,  in  ex.  (1). 

Notice,  in  each  case,  that  this  process  permits  each  person  to  make  or 
to  receive  settlement  in  the  money  of  his  own  country. 

ORAL   EXERCISE. 

1.  In  what  coinage  would  an  American  importer  order  of  a  French 
exporter?  If  he  bought  exchange,  with  which  to  make  settlement,  in 
what  coinage  would  it  read?  With  what  money  would  he  make  settlement 
for  the  draft? 


DOMESTIC  AND  FOREIGN  EXCHANGE.  373 

2.  An  English  dealer  orders  goods  from  the  catalogue  of  an  American 
manufacturer.  He  finds  prices  quoted  in  what  coinage?  If  he  pays  by 
bank  draft,  the  draft  is  made  out  in  what  money?  If  he  is  drawn  upon, 
in  settlement,  the  draft  is  made  out  in  what  money?  If  the  manufacturer 
sells  the  draft  in  New  York  he  will  receive  money  of  what  country? 

How  would  you  determine: 

3.  The  cost  of  a  draft  on  Paris,  at  5.19? 

4.  The  cost  of  a  draft  on  London  at  4.88? 

5.  The  cost  of  a  draft  on  Berlm  at  .958? 

6.  The  proceeds  from  the  sale  of  a  draft  on  Lyons  at  5.185? 

7.  The  proceeds  from  the  sale  of  a  draft  on  Amsterdam  at  40$%? 

8.  The  proceeds  of  a  draft  on  Berlin  at  96i? 

9.  Which  are  the  higher  rates,  5.19  or  5.20?    96  or  95^? 

EXERCISE. 

(For  Mint  exchange  rates,  see  page  281.) 

1.  What  is  the  face  of  a  draft  that  an  English  exporter  should  draw 
on  a  New  York  customer,  at  mint  par,  to  cover  the  purchase  of  240  yd. 
broadcloth  at  0/14/6  per  yd.?  If  the  exporter  sells  this  draft  in  the  market 
for  4.87,  what  proceeds  does  it  yield  him? 

Compute  the  cost  of  drafts  on: 

2.  Paris,  for  9700  francs  at  5.18 

3.  Berlin,  for  12,500  M.  at  .962. 

4.  Glasgow,  for  £161  4s.,  at  4.87|. 

5.  Amsterdam,  for  752  g.,  at  .405. 

6.  Rome,  for  768  L.  at  .238. 

7.  Geneva,  for  1284.6  fr.,  at  5.18|,  brok.  1/8%. 

8.  Is  the  balance  of  trade  probably  for  or  against  this  country  if  ex- 
change is  quoted  at  5.2,  4.88,  .412? 

Exchange  Rates. 
(From  a  daily  newspaper.) 

Sight.  60  days 

Cable  transfers 4.87?  

Bankers'  sterling 4.87  4.85^ 

Commercial  bills 4.84^ 

Francs 5.161  5.171 

Reichsmarks ; " 95|  .95i 

Guilders 40|  .40i 


374  BUSINESS  ARITHMETIC. 

Note.     Use  the  newspaper  quotations  on  page  373  in  the  following: 

9.  James  Bros.  &  Co.  purchase  an  invoice  of  £2000  worth  of  woolens, 
paying  by  60-day  bank  draft.     Cost  of  draft? 

10.  How  large  drafts  would  $500  buy  at  each  of  the  above  quotations? 

11.  A.  B.  Norton  drew  on  a  Paris  customer  a  draft  for  12,500  fr.  and 
sold  it  at  1/8%  commission,  receiving  what  amount? 

12.  What  does  it  cost  to  cable  an  agent  £120? 

13.  B.  C  Baker  shipped  S.  Tournier,  Paris,  5000  bu.  wheat,  invoiced 
at  4.8  fr.  drawing  a  30  day  draft  on  the  purchaser.  This  draft,  accom- 
panied by  bill  of  lading  and  insurance  certificate,  he  sold  to  Mason  and 
Dean,  private  bankers,  at  5.19,  and  should  receive  what  sum? 

14.  Mason  &  Dean  sold  the  draft  just  mentioned  to  Robert  Bros.,  at 
5.18,  receiving  $ — ,  and  making  a  profit  of  $ — 

15.  Robert  Bros,  mailed  the  same  draft,  in  settlement  of  account,  to  a 
Paris  customer,  who  discounted  it  at  a  bank  at  1%,  receiving  —  fr.  The 
bank  presented  the  draft  to  Tournier(ex.  13)  who  was  allowed  to  settle  for 
it  instead  of  accepting  it,  at  1/2%  discount.  He  then  received  the  bill 
of  lading  that  entitled  him  to  the  wheat. 


CHAPTER  XLI. 

DEPRECIATION. 

Depreciation  is  the  loss  of  value  of  any  property  due  to  weaj*, 
new  inventions,  exhaustion  of  product,  or  other  causes.  It 
is  the  difference  between  original  value  and  marketable  value 
at  time  of  disuse. 

Illustrations.  (1)  In  spite  of  repairs,  a  factory  or  piece  of  machinery- 
deteriorates  to  the  point  where  it  must  be  discarded  or  rebuilt.  This 
deterioration  is  depreciation.  (2)  New  inventions  or  improvements  may- 
make  old  machinery  valueless.  (3)  A  lessening  of  demand  for  a  product 
may  cause  the  disuse  of  machinery,  and  a  consequent  lessening  in  value. 
(4)  A  mine  deteriorates  from  exhaustion  of  its  ore,  etc. 

To  avoid  the  necessity  of  meeting  the  entire  cost  or  replace- 
ment in  any  one  year,  it  is  customary  in  large  concerns,  to 
allow  for  depreciation,  setting  aside  each  year  during  the 
probable  useful  life  of  the  property,  sufficient  funds  to  equal 
the  cost  of  replacement  when  necessary.  The  sum  set  aside 
is  largely  a  matter  of  scientific  guess-work  based  on  past 
experience.  It  is  considered  a  running  expense  and  may 
amount  to  30%  of  such  expenses.  Allowing  for  individual 
variation,  the  following  table  shows  common  rates. 

Annual  Rates  op  Depreciation. 

Modern  fireproof  steel  and  tile  factories 4%    to    5%. 

Steel  construction,  partly  fireproof 6%. 

Warehouses,  modem,  fireproof 2^%  to    4%. 

General  freehold  buildings,  office  and  general  use 1|%  to    3%. 

Cheaply  constructed  building,  according  to  use 8%    to  15%. 

Fixtures  of  power  and  lighting  plants 12%    to  15%. 

General  fixtures  of  buildings,  other  than  machinery 6%    to  15%. 

Machinery  (average  11§%) 8%    to  15%. 

Horses,  work 15%    to  25%. 

375 


376  BUSINESS  ARITHMETIC. 

Annual  amounts  of  depreciation  are  computed  (1)  as  a  fixed 
proportion  of  the  original  value;  (2)  as  a  fixed  per  cent,  of 
reduced  values;  or  (3)  by  the  sinking  fund  method. 

1.     Based  on  Original  Value. 

Illustration.  A  machine  in  a  cotton  mill,  costing  $560,  has  an 
average  working  life  of  10  years,  when  it  is  disposed  of  for  $120.  Determine 
the  annual  allowance  for  depreciation. 

Solution.  Depreciation  =  Original  cost  — disuse  value  =  $560  —  $120 
=  $440.  Since  this  must  be  covered  in  10  years  the  annual  depreciation  is 
1/10  of  $440,  or  $44. 

EXERCISE. 

1.  Compute  the  annual  depreciation  at  2f  %,  on  a  fireproof  warehouse 
costing  originally  $364,200. 

2.  Compute  the  annual  depreciation  on  $340,200  worth  of  fixed  ma- 
chinery, at  12%,  and  on  tools  originally  worth  $3640,  at  18%. 

3.  A  department  store  owner  has  40  dehvery  horses  originally  costing 
$5680.     He  reckons  the  depreciation,  at  22^%,  as  $ . 

4.  A  boiler,  costing  $520,  is  estimated  to  have  a  life  of  12  years,  and 
then  to  be  worth  $60.  Estimate  annual  depreciation  and  rate  of  depre- 
ciation. 

5.  A  contractor  employs  a  $14,500  dredge  in  contract  work.  In  com- 
puting cost  of  work,  as  a  basis  for  bids,  he  includes  interest  on  the  cost  of 
the  dredge  and  depreciation.  He  reckons  interest  at  6%  per  annum  and 
depreciation  1|%  per  month.  What  should  he"  allow  for  interest  and 
depreciation  on  a  contract  that  he  estimates  will  take  7  5  months? 

II.    Based  on  Reduced  Values. 

With  machinery,  for  example,  repairs  grow  heavier  with  age.  Hence, 
with  uniform  depreciation  and  increasing  repairs,  the  working  cost  is 
heaviest  when  the  machine  is  least  useful.  To  equalize  the  yearly  cost 
therefore,  depreciation  is  often  heavier  when  repairs  are  light,  decreasing 
as  repairs  grow  heavy.  Depreciation  of  this  type  is  often  expressed  as 
a  fixed  per  cent,  of  the  net  value  of  the  previous  year. 

Illustration.  Compute  the  third  year's  depreciation,  at  10%  on 
reduced  values,  of  machinery  costing  $900. 


DEPRECIATION.  377 


Solution, 


$900       Orig.  value 

90       Depreciation  at  10%. 
810       Net  value,  end  1st  year. 
81       Dep.  10%. 
729       Net  value  2d  year. 
72.90  Dep.  10%,  for  3d  year.     Ans. 
Note.     If  term  and  rate  of  depreciation  are  known,  tables  similar  to  the 
following  may  be  computed  for  shop  use. 

Table  of  Depreciation  per  $100  op  Original  Value. 


Term  Year. 

2% 

3% 

4% 

5% 

10% 

15% 

20% 

lyr. 

2.00 

3.00 

4.00 

5.00 

10.00 

15.00 

20.00 

2 

1.96 

2.91 

3.84 

4.75 

9.00 

12.75 

16.00 

3 

1.92 

2.82 

3.69 

4.51 

8.10 

10.84 

12.80 

4 

1.88 

2.74 

3.54 

4.29 

7.29 

9.21 

10.24 

5 

1.85 

2.66 

3.40 

4.07 

6.56 

7.83 

8.19 

6 

1.81 

2.58 

3.26 

3.87 

5.90 

6.66 

6.55 

7 

1.77 

2.50 

3.14 

3.68 

5.31 

5.66 

5.24 

8 

1.73 

2.42 

3.00 

3.49 

4.78 

4.81 

4.19 

EXERCISE. 

1.  Compute  the  depreciation  for  the  sixth  year,  at  8%  on  reduced 
balances,  on  fixtures  originally  worth  $8240. 

2.  A  factory  costing  $82,600.00  is  five  years  old.  What  is  its  inventory 
value,  if  depreciation  is  reckoned  on  reduced  balances  at  4|%? 

By  Table. 

3.  Reckon  the  depreciation  for  the  eighth  year,  on  machinery  worth 
originally  $6525,  if  the  depreciation  rate  is  10%. 

4.  What  is  the  total  loss  of  value  in  seven  years,  of  a  boiler  costing 
$1800,  and  depreciated  15%  annually? 

5.  Compute  the  inventory  value  at  the  end  of  8  years,  of  an  equipment 
of  cotton  mill  machinery,  costing  originally  $38,250,  and  depreciated  at 
20%  annually. 

6.  Determine  a  rate  of  depreciation  for  machinery  that  is  expected  to 
depreciate  from  $900  to  $630  in  7  years. 

III.     Sinking  Funds. 
Depreciation  is  sometimes  provided  for  by  establishing  a  sinking  fund, 
that  is  a  fund  which  is  invested  at  compound  interest  at  the  rate  necessary 


378 


BUSINESS  ARITHMETIC. 


to  provide  for  renewals.  Frequently  this  fund  is  based  on  annual  invest- 
ments of  equal  amounts.  Business  establishments  seldom  establish  a  sink- 
ing fund,  since,  when  prosperous,  funds  earn  a  higher  rate  by  being  con- 
tinued in  the  business,  than  by  outside  investment. 

Sinking  Fund  Table. 
Shxywing  Accumulations  oi  Compound  Interest  of  Annual   Investments  of 

One  Dollar. 


Year 

At  4^ 

At  5^ 

At  6^ 

Year 

At45< 

At6?t 

At6)t 

1 

1.040000 

1.050000 

1.060000 

11 

14.025805 

14.917127 

15.869941 

2 

2.121600 

2.152500 

2.183600 

12 

15.626838 

16.712983 

17.882138 

3 

3.246464 

3.310125 

3.374616 

13 

17.291911 

18.598632 

20.015066 

4 

4.416323 

4.525631 

4.637093 

14 

19.023588 

20.578564 

22.275970 

5 

5.632975 

5.801931 

5.975319 

15 

20.824531 

22.657492 

24.672528 

6 

6.898294 

7.142008 

7.393838 

16 

22.697512 

24.840366 

27.212880 

7 

8.214226 

8.549109 

8.897468 

17 

24.645413 

27.132385 

29.905653 

8 

9.582795 

10.026564 

10.491316 

18 

26.671229 

29.539004 

32.759992 

9 

11.006107 

11.577893 

12.180795 

19 

28.778079 

32.065954 

35.785591 

10 

12.486351 

13.206787 

13.971643 

20 

30.969202 

34.719252 

38.992727 

EXERCISE. 

1.  What  is  the  sum  that  must  be  invested  annually  in  a  4%  sinking 
fund  to  provide  for  replacing  a  $12,000  equipment  of  machinery  in  10 
years,  if  the  estimated  value  of  the  equipment  at  the  end  of  the  period  is 
$1950? 

2.  What  allowance  for  depreciation  should  be  made  annually  for 
machinery  that  is  expected  to  deteriorate  in  value  $6750  in  5  years,  when 
it  must  be  replaced?     The  sinking  fund  is  invested  at  6%.     What  at  5%? 

3.  A  certain  factory  building  is  estimated  to  have  a  life  of  20  years. 
It  is  built  at  a  cost  of  $46,000.  Estimate  uniform  annual  depreciation. 
Compute  annual  allowance  for  sinking  fund  at  6%. 


CHAPTER  XLII. 
COST-KEEPING. 

Knowledge  of  the  exact  cost  of  the  articles  that  he  makes  is 
of  vital  importance  to  a  manufacturer.  Owing  to  slight  margins 
of  profit,  carelessness  in  estimating  materials  and  labor,  or 
needless  waste,  may  lead  to  large  losses. 

The  determining  of  the  exact  cost  of  an  article,  or  piece  of 
work,  is  termed  cost-keeping.  It  necessitates  (1)  the  exact 
measurement  of  the  material  used  and  wasted,  (2)  the  com- 
puting of  the  time  spent  on  the  work  by  each  workman, 
and  (3)  the  computation  of  the  exact  cost  of  material  and 
labor.  The  labor  cost  is  often  greater  and  more  variable  than 
the  material  cost,  and  is  considered  the  vital  element  in  cost- 
keeping. 

Note.  The  computation  of  the  cost  of  production  is  often  exceedingly- 
involved,  and  varies  with  each  class  of  work,  but  the  fundamental  principles 
are  very  simple  and  are  worth  considering,  since  the  subject  is  becoming 
a  matter  of  general  discussion,  and  of  frequent  mention  in  current  pul> 
lications. 

Illustration.  A  printing  office  receives  an  order  for  2000  booklets 
of  a  specified  character.  At  once  a  dupHcate  docket  (record  sheet)  is 
prepared  (page  380)  to  show  full  particulars  of  cost.  The  original  docket 
accompanies  the  work  from  start  to  finish,  quantities  and  costs  being 
entered  at  each  stage,  as  determined,  the  form  being  completed  when 
the  work  is  finished.  Each  workman  keeps  his  separate  ticket,  and  marks 
on  it  the  hours  that  he  spends  on  each  job.  These  tickets  are  turned  into 
the  office  at  night,  and  the  time  and  labor  costs  transferred  to  the  dupli- 
cate docket  here  on  file.  Labor  costs  are  also  summed  up  and  entered  on 
the  original  docket  as  it  passes  from  hand  to  hand.  The  duplicates  thus 
serve  as  checks.  Each  job,  however,  must  bear  its  share  of  the  expense 
of  the  office  force,  foremen,  etc.  (the  non-productive  labor),  and  its  share  of 
rent,  wear  on  machinery,  lighting,  etc.  (biu-dens) .  This  expense,  after  long 
experience,  is  expressed  as  a  per  cent,  of  direct  cost,  or  of  labor  cost  only. 
To  this,  if  no  advance  price  has  been  quoted,  is  added  the  per  cent,  of 
profit.  The  completed  cost  cards  are  filed  to  serve  as  a  basis  for  estimating 
on  future  work.     Errors,  carelessness,  and  excessive  cost  of  labor,  may  also 

379 


380 


BUSINESS  ARITHMETIC. 


be  traced  from  them.     Many  improvements  are  suggested  by  a  study  of 
cost  cards. 


1 

JOB   WORK    DOCKET 

fa/m^  ^  jV<yyl(yn         ^om.  ^9,  i/9i/2 

oue   ^'^/26 

726  ^^t..  jv.  ir. 

DESCRIPTION 

CHARGFD   ^^<^U.   2 G ,    /^/i*                                                                                            ^MOMK-T        S 75.00 

STOCK    SUPPLIED    WITH    DOCKET 

ITEMS 

TOTALS 

'/O 

OS 

i^00J2S      ,^.   /^  ^jc^^.^                  %^0.00 

F 

F 

R      Q         S                                                       $ 

R      Q        S                                                       $ 

COVERS                                   $ 

PRINTING 
t                    COST 

1- 

COMPOSITION 

/J? 

08 

PRESSWORK 

7 

24 

INK 

/ 

€0 

F 

ROM      WORK 

§  i 

H     Z 

LABOR 

A 

48 

MAIL  AND  POSTAGE 

5 

STOCK 

2 

00 

F 

OUTSIDE   WORK 
COST 

2    ^Ua^oi.  l/.AO 

2 

80 

TOTAL    COST 

F 

FOREMAN  PRINTERV                                                                                              FOREMAN  BINDERY 

EXERCISE. 

1.  Extend  the  job  work  docket,  shown  above. 

2.  Compute  the  cost  of  a  box  requiring  2  pc.  wood,  1X6X8;  2  pc. 
1/2X6X12,  and  2  pc.  1/2X8X12;  waste  4%.  Cost  of  material,  2fc  per 
ft.;  labor,  sawing,  etc.,  $.006  per  ft.;  assembling  $.025  per  bx.;  indirect 

3,  20%  of  labor.     (Reckon  values  to  four  decimal  places.) 


COST-KEEPING. 


381 


3.     Complete  the  following  card. 

NO.    PIECES                                   NAME     OF    ARTICLE 

ff^O                             ^_  41=  €78 

SHOP    ORDER 

730e 

ORDERED                                                                                 ^^^^ 

J^a9i.  S7.  1/9/2                                         Ja<n..  S9 

FINISHED 

DEPARTMENT 

OPER. 

HOURS 

RATE 

COST 

MATERIAL 

COST 

9 

■/S 

? 

\450'WrJn/'  9.00 

/3 

05 

// 

C/ncr^iin^ 

/2 

./2 

:? 

.^//e^ 

4 

3S 

// 

CUc^icU'na. 

/e 

.09 

y 

^/J^  ^^^^ 

5 

99 

^nA/iin^ 

^aic^g. 

^2^/2 

.20 

p 

iiUe 

87 

Jcc^ftiiA.  'a 

1/3 

.25 

? 

0>o/is/,'a, 

/5% 

.30 

.^ 

MATERIAL  COST 

:? 

*SHOP    BURDEN 

? 

COST  PER  PC.     MAT. 

? 

COST  PER  PC.     LABOR 

F 

LABOR  COST — 630  PC. 

p 

/ 

COST  PER  PIECE 

F 

- 

'/2 


Woo/  /a^^o-^c 


4.     Complete  the  following  card. 

ARTICL.      3000 

FORSBERG   STEEL 

DATE 

CO. 

t/f^  <ffOcAels                           „„ 

^—  673^ 

QUANTITY 

OPERATION 

TIME 

RATE 
PER  100 

NO.  OF 
OPERATOR 

3000 

PMoc/t- 

20A 

.45 

72/ 

F 

3000 

PJi/^^l 

32A 

.52 

483 

F 

3000 

.■ri/yi/A/l. 

^5A 

.90 

52/ 

p 

^oal  /8  i    /teiK  A^. 

p  /x. 

F 

7000*     Alee^ 

2^/ji^ 

9 

^^t^^ntn.&it ,           ■/23  A,^. 

/./8 

F 

0>,.^.J.                       7  ^^. 

.80 

F 

.^...                          7  A*. 

.25 

F 

.^/^y 

F 

-»-  35  Vo 

p 

^^^ 

F 

F 

382  BUSINESS  ARITHMETIC. 

Note.  In  the  form  just  shown,  operations  axe  paid  for  on  the  "piece" 
plan,  and  coal  and  use  of  machinery  are  charged  by  the  hour,  previous  ex- 
periments having  determined  what  is  a  fair  charge. 

The  construction  engineer,  or  contractor,  estimates  costs 
per  unit  of  finished  work,  or  per  day,  and  holds  these  unit  costs 
as  a  basis  for  future  estimating. 

EXERCISE. 

1.  A  gang  of  seventeen  men,  working  on  a  concrete  construction, 
averaged  46  cu.  yd.  of  concrete  per  day.  Estimate  labor  costs  per  day 
and  per  cubic  yard,  based  on  the  following  organization: 

Per  day.  Per  cu.  yd. 

5  men  loading  barrows,       at  $1.75  per  day,  $ . —  $ . — 

9  men  mixing  and  placing,  at  $1.75  per  day,    . —         . — 

2  men  tamping,  at  $1.75  per  day,    . —         . — 

1  foreman,  3.00  . — 

2.  A  gang  of  men  averaged  42  lin.  ft.  of  concrete  conduit  (3/4  cu.  yd. 
per  lin.  ft.)  per  day.     Estimate  the  labor  cost  per  day  and  per  cu.  yd. 

12  men  mixing,  at $1.80 

4  helpers ". 1.50 

6  loading  stone  and  sand 1.50 

6  wheeling 1.50 

8  pouring  concrete 1.80 

2  supplying  water 1.50 

2  placing  metal 2.35 

1  sub-foreman 2.75 

1  foreman ' 3.50 

3.  Estimate  the  unit  cost  per  cubic  yard  for  260  cu.  yd.  of  concrete, 
laid  in  seven  8-hour  days. 

Materials  and  Labor. 
0.85  bbl.  cement,  per  cu.  yd.,  at  $2.90  per  bbl. 

.25  cu.  yd.  sand,  per  cu.  yd.,  at  $1.12. 

.50  cu.  yd.  broken  stone,  per  cu.  yd.,  at  $.80. 

.40  cu.  yd.  rubble,  at  $.55. 
Water,  for  the  entire  quantity,  $8.00. 
Labor,  9  men,  at  18c  per  hour,  each. 
Foreman,  $3.60  per  day. 


CHAPTER  XLIII. 
BIDS  AND  ESTIMATES. 

Those  desiring  work  done,  or  material  supplied,  often 
advertise  for  seale'd  proposals,  or  bids,  from  such  contractors 
as  care  to  meet  their  requirements.  These  bids  are  based  on 
specifications  supplied  by  the  advertiser,  which  state  in  detail 
the  work  to  be  accomplished,  liow  it  is  to  be  done,  the  time  of 
completion,  the  character  of  material  to  be  used,  the  conditions 
as  to  the  acceptance  of  the  completed  work,  payments,  etc. 

The  prospective  bidder  makes  an  estimate  of  the  cost  of  the 
work,  allows  for  his  profit  and  prepares  his  bid  in  due  form, 
usually  on  a  blank  supplied  with  the  specifications.  The 
sealed  proposals  are  opened,  frequently  in  the  presence  of  all 
bidders  who  care  to  attend,  on  a  fixed  date,  and  the  contract 
awarded.  While  the  lowest  bidder,  if  .reliable,  frequently 
secures  the  contract  for  the  work,  parts  of  the  work  may  be 
given  to  different  contractors.  The  proposal  often  names 
prices  for  parts  of  the  work,  or  for  the  work  as  a  whole. 
Usually  the  bid  must  be  accompanied  by  a  certified  check 
for  a  certain  percentage  of  the  amount  bid,  as  security  for  the 
acceptance  and  performance  of  contract,  if  tendered;  or  a 
bond  must  be  given  as  security. 

Model  Proposal  for  a  Gravel  Road, 
proposal. 

To  THE  Board  of  Chosen  Freeholders,  County  of 

AND  State  of  New  Jersey: 
Gentlemen — The  undersigned  hereby  declare. .  .that. . .  .he. . .  .ha. . . . 
carefully  examined  the  annexed  specifications  and  the  drawings  therein 
referred  to,  and  will  provide  all  necessary  machinery,  tools,  apparatus 
and  other  means  of  construction,  and  do  all  the  work  and  furnish  all  the 
material  called  for  by  said  specifications  in  the  manner  prescribed  by  the 

383 


384  BUSINESS  ARITHMETIC. 

specifications  and  the  requirements  of  the  Engineer  and  Supervisor  under 
them,  for  the  following  prices: 

(1)  Price  per  cubic  yard  for  earth  excavation,  without  classification,  as 

per  plans  and  cross-sections,  throughout  the  length  and  width  of 
the  road per  cubic  yard. 

(2)  Price  per  acre  for  grubbing  and  removing  objectionable  material  from 

sidewalks per  acre. 

(3)  Price  per  lineal  foot  for  completed  tile  drain per  lineal  foot. 

(4)  Price  per  cubic  yard  for  compacted  gravel  as  specified 

per  cubic  yard. 

(5)  Price  per  cubic  yard  for  carting  gravel  more  than  one  half  mile  and 

each  additional  half-mile  or  fraction  thereof 

per  cubic  yard. 

(6)  Price  per  cubic  yard  for  stripping  or  removing  earth  from  top  of 

gravel  bed per  cubic  yard. 

{7)  Price  per  square  yard  for  each  ordered  inch  in  depth  in  excess  of 

thickness  named per  square  yard. 

(8)  Price  (lump)  for  the  whole  road  complete,  according  to  the  specifica- 
tions and  plans  prepared  by  the  Engineer 

Accompanying  this  proposal  is  a  certified  check  for  the  sum  of  one 
thousand  ($1,000)  dollars,  payable  to  the  order  of  the  Director  of  the  Board 

of  Chosen  Freeholders  of .County;  which 

check  is  to  be  forfeited  as  hquidated  damages,  if,  in  case  this  proposal  is 
accepted,  the  undersigned  shall  fail  to  execute  a  contract  with  said  Board 
of  Chosen  Freeholders,  under  the  conditions  of  this  proposal,  within  the 
time  provided  for  by  the  foregoing  advertisement  for  proposals;  other- 
wise said  check  is  to  be  returned  to  the  undersigned. 

Signed 

Address 

N.  J., 

In  preparing  his  bid  the  contractor  must  allow  for  such 

things  as: 

1.  Preparatory  expense.     Cost  of  preparing  for  the  real  work,  such  as 
setting  up  working  machinery,  clearing  ground,  etc. 

2.  Plant  expense  and  supplies.     Cost  of  maintaining  tools,  machinery, 
worksheds,  etc.     Supphes  for  equipment.     Depreciation  allowance. 

3.  Materials.     Those   used  in   the  real   construction.     Freight   and 
cartage. 

4.  Labor.     Skilled  and  unskilled  workmen. 


BIDS  AND  ESTIMATES.  385 

5.  Superintendence  and  general  expense.  Wages  of  firemen,  time- 
keepers, superintendent,  clerical  force.  Supplies  for  their  use.  This 
expense  varies  from  5%  to  15%  of  labor. 

6.  Contingencies.  Allowance  for  unforeseen  expense,  such  as  delays 
from  bad  weather,  labor  troubles,  accidents,  rise  in  price  of  materials- 
Often  in  excess  of  10%  of  cost. 

7.  Profit  allowance.  At  times  a  fixed  per  cent,  of  cost.  At  times, 
one  rate  on  cost  of  materials,  say  10%  to  15%,  and  a  second  rate,  say  15% 
to  25%  on  labor.  These  rates  are  often  made  high  enough  to  cover  probable 
contingencies. 

Note.  Estimates  of  cost  are  based  on  current  prices  of  labor,  freight, 
materials,  etc. :  estimates  of  quantities,  largely  on  past  experience.  Often 
the  unit  cost,  as  obtained  by  cost-keeping  methods,  is  the  basis  for  an 
estimate.  The  exercises  that  follow  illustrate  a  few  principles  of  the 
subject. 

ORAL    EXERCISE. 

1.  A  bidder  for  a  contract  for  office  supplies  believes  that  he  can  secure 
the  material  for  $3800.  He  reckons  freight  at  $87.50;  labor  at  $32.50. 
Estimate  his  bid  on  a  profit  allowance  of  20%. 

2.  Compute  a  bidder's  profit  of  15%  on  labor  estimated  to  cost  $38o. 

3.  Compute  a  bidder's  allowance  for  12^%  profits  on  materials  costing 
him  $1840. 

4.  A  contractor  estimates  the  gross  cost  of  a  prospective  structure  at 
$28,500.  If  he  allows  20%  for  profits  and  contingencies,  what  should  his 
bid  be? 

EXERCISE. 

1.  A  contractor  estimates  the  cost  of  materials,  plant,  etc.,  for  a 
certain  work,  at  $46,758,  and  of  labor  at  $39,200.  He  adds  10%  to  labor 
for  contingencies.  He  then  reckons  a  profit  of  20%  on  labor  and  12^% 
on  materials.     What  is  the  amount  of  his  bid? 

2.  Gross  estimated  cost  of  materials,  $15,286.50;  plant  expense,  $3495; 
labor,  $7296.40.  Profit  on  materials,  15%;  on  labor,  25%.  Amount  of 
bid? 

3.  Reckon  quantities  of  material,  cost  of  labor  and  arnqjint  of  bid,  for 
a  contract  of  260  squares  of  slate  roofing,  basing  values  on  these  figures: 
(1)  Unit  cost  per  square,  214  pc.  slate,  $5.60;  freight,  $1.40;  loading  and 
hauling,  35c;  slate  waste,  2%;  16  lb.  sheathing  paper,  50c;  1  lb.  nails,  5c; 
21  lb.  nails,  galvanized  9c;  labor,  $1.70.  (2)  Superintendence,  10%. 
(3)  Labor  contingency,  10%.  (4)  Profit,  on  net  cost  of  labor,  20%;  on 
materials,  15%. 

26 


386  BUSINESS  ARITHMETIC. 

4.  Compute  the  face  of  bid  on  14,365  cu.  yd.  concrete  foundation, 
based  on  a  unit  cost  per  cu.  yd.,  as  follows:  Stone,  $1.15;  gravel  S.14; 
sand,  $.08;  cement,  $1.36;  labor,  $1.05;  superintendence,  $.05.  Profits 
on  labor,  25%;  on  materials,  15%. 

5.  Compute  bid  (ex.  4),  allowing  a  straight  profit  of  20%  on  cost  per 
unit. 

•  6.  A  contractor  agrees  to  cut  and  supply  60,000  ft.,  B.  M.,  of  3" 
plank,  basing  this  bid  on  the  following  cost  per  unit  M  ft.:  stumpage, 
$1.75;  cutting,  $1  40;  skidding,  $.75;  sawing,  $2.40.  Compute  his  bid, 
allowing  25%  for  profit  and  contingencies. 


CHAPTER  XLIV. 

PARTITIVE    PROPORTION    AND    PARTNERSHIP. 

Partitive  proportion,  one  phase  of  ratio,  is  the  process  of 
dividing  a  number  into  parts  proportional  to  two  or  more 
given  numbers. 

ORAL    EXERCISE. 

1.  Compare  4  and  6. 

2.  A's  12  shares  compare  how  with  B's  4  shares?  Together  they 
have  how  many  shares? 

3.  Divide  $320  into  parts  proportional  to  2  and  3. 

Suggestion.     2+3=5  parts. 
$320  =  5  parts. 

1  part  =  $64. 

2  "      = ? 

3  "      = ? 

4.  Divide  $7200  in  the  ratio  of  5  to  7. 

5.  Divide  1200  yd.  into  two  lots  having  the  ratio  3/5. 

6.  Divide  6300  lb.  into  parts  proportional  to  1,  2  and  3. 

7.  Divide  $1800  into  parts  proportional  to  1/4,  1/3,  and  1/6. 

8.  Divide  $600  between  A  and  B,  so  that  A  receives  3  times  as  much 
as  B. 

9.  Two  dealers  unite  in  ordering  a  carload  of  40  tons  of  soft  coal. 
The  freight  bill  is  $24.  If  the  first  dealer,  who  bought  15  tons,  settles  the 
bill,  what  should  the  second  dealer  pay  him? 

10.  A  piece  of  freight  is  carried  200  miles  on  one  railway,  and  150 
miles  on  a  second,  for  a  total  charge  of  $35.00.  What  should  each  railway 
receive? 

A  partnership  is  an  association  of  two  or  more  persons  for 
the  purpose  of  conducting  a  business  in  common,  and  of 
sharing  in  its  profits  or  losses.  Partnerships  are  formed  under 
verbal  or  written  agreements.     The  latter  usually  include  a 

387 


388  BUSINESS  ARITHMETIC. 

statement  of  the  conditions  concerning  investments,  with- 
drawals, and  the  distribution  of  profits. 

Collectively,  the  persons  forming  a  partnership  are  termed 
a  firm,  a  house,  or  a  company.  Individually,  they  are  known 
as  partners.  Partnerships  may  be  conducted  under  fictitious 
names,  under  the  names  of  the  partners,  or  under  the  name 
of  one  or  more  partners  "  and  Co. "  Partners  are  real  or  ostens- 
ible, if  they  are  really  partners  and  are  known  to  the  public  as 
such.  They  are  silent  partners  if  their  connection  with  the 
firm  is  not  known  to  the  public;  nominal,  if  they  make  no 
investment  and  have  no  share  in  the  proceeds  of  the  business; 
or  limited  if  they  have  only  a  partial  legal  responsibility  for 
the  debts  of  the  firm.  Real  and  nominal  partners  are  re- 
sponsible for  all  liabilities  of  their  firm.  Silent  partners  when 
their  connection  becomes  known  must  assume  their  share  of 
responsibility  for  the  liabilities  of  the  firm.  Limited  and  silent 
partnerships  are  not  allowed  in  some  states;  in  others  it  is 
often  required  that  at  least  one  partner  shall  be  real. 

The  capital  of  the  partnership  is  the  property  invested  in 
the  business.  It  includes  money,  any  tangible  form  of  real 
or  personal  property,  labor,  good  will,  etc. 

Note.  Good  will  is  the  term  applied  to  the  value  of  past  custom 
of  an  old  established  place  of  business,  in  bringing  future  trade,  or  of 
confidence  in  a  partner  who  "lends  his  name,"  etc.  Such  a  partner  is 
often  credited  with  good  will  as  an  investment,  at  some  money  value,  on 
which  his  share  of  profits  may  be  based. 

The  net  capital  or  present  worth  of  a  firm  at  any  time  is  the 

excess  of  resources  or  assets  (property  owned  and  debts  due) 

over  liabilities  (debts  owed).     If  liabilities  are  in  excess  there 

is  net  insolvency, 

FOR    DISCUSSION. 

Compare  partnerships  and  stock  companies  as  to: 

1.  Method  of  organization.  5.    Effect  of  death  of  partner  or 

2.  Method  of  investing.  stockholder. 

3.  Withdrawal  from  business.  6.     Government  and  management. 

4.  Sale  of  interest  in  business.  7.    Individual    responsibility    for 

debts. 


PARTITIVE  PROPORTION  AND   PARTNERSHIP.    389 

ORAL    EXERCISE. 

1.  Total  gains  —  total  losses  =  ? 

2.  Total  losses  —  total  gains  =  ? 

3.  Resources  =fc  ?  =  net  capital. 

4.  Original  capital  +  net  gain  =  ? 

5.  Original  capital  —  net  loss  =  ? 

6.  Resources  —  liabilities  =  ? .     How  check? 

7.  May  a  firm  gain  during  a  period  and  be  insolvent  at  its  close? 

EXERCISE. 
Supply  missing  words  or  values.     Solve  mentally  if  possible. 

1.  Gains  $2400;  losses  $560;  net -,  $ . 

2.  Gains,  $2150;  losses  $2680;  net ,  $ . 


3.     Capital  at  commencing  $16000;  at  close,  $18500;  net =  $- 


4.  Original  capital,  $5000;  gains,  $1250;  losses  $640;  net  capital  at 
close,  $ . 

5.  Net  insolvency,  Jan.  1,  1907,  $2500;  Jan.  1,  1908— Resources,  $5260; 
liabilities  $5000.     Result  of  business  for  the  year  a of  $ . 

6.  A  &  Co.  invested  $12,000,  and  a  year  later  had  cash,  $2000;  mdse., 
$10,500;  real  estate  $2500,  but  owed  debts  of  $2000.  A  &  Co.'s  net 
capital  then  was  $ . 

7.  Jan.  1,  1912.  Net  capital  $5000.  Jan.  1,  1913,  gross  gains  during 
year  $3000;  gross  losses,  $1200.  Present  resources  $8000;  liabiUties  $900. 
Prove  the  existence  of  an  error  in  these  figiu-es. 

Distribution  of  Profits  and  Assessment*  of  Losses. 

In  partnerships,  profits  are  distributed  and  losses  met  in 
proportions  agreed  upon.  These  may  be  arbitrary  or  graded 
for  special  service,  or  based  on  amounts  of  investments;  or  a 
combination  of  these  factors. 

I.    EQUAL  DIVISION   OF  PROFITS. 
ORAL    EXERCISE. 

1.  Divide  $726  profits  equally  between  Adams  and  Brown. 

2.  A's  investment  was  $4000  and  D's  $6000.  They  share  equally  a 
profit  of  $800,  but  D  withdraws  $100  of  his  share.  The  net  capital  of  the 
firm  is  then  $  ?,  and  that  of  D,  $  ? 


390  BUSINESS  ARITHMETIC. 

3.  A  invests  $2000,  B  $4000,  and  C  $3000.  Determine  their  present 
worths,  after  they  have  shared  equally  a  loss  of  $960. 

EXERCISE. 

1.  July  1,  1908.     Investments:  A,  $12,600;  B,  $14,800;  C,  $9,650. 
Jan.  1,  1909.     Withdrawals;  A,  $1200;  B,  $400. 

Jan.  1,  1909.     Gains:     On  mdse.,  $7260.50;  on  real  estate  $1260. 
Losses:  Expense,  $3126.50;  fuel  and  light  $568.40;  taxes  $165. 
Find  the  present  worth,  if  profits  are  shared  equally. 

2.  On  Jan.  1,  Evans  invested  $12,000,  Brown  $8000  and  Thompson 
$15,000  in  the  real  estate  business.  A  year  later  their  resources  were: 
Cash  $7290;  accounts  and  notes  receivable,  $11,360;  real  estate  $26,750, 
while  they  owed  notes  for  $3296.50  and  taxes  of  $561.30.  Find  the  present 
worth  of  each  member. 

II.     PROFITS  SHARED  IN  ARBITRARY  PROPORTIONS. 
ORAL    EXERCISE. 

1.  Divide  $792  between  A  and  B  in  the  ratio  of  3  to  1. 

2.  A  receives  2/3  of  the  profits  of  a  business  in  which  he  invested  1/4 
of  the  capital  of  $12,000.     Find  his  present  worth  if  the  profits  are  $900. 

EXERCISE. 

1.  Crown  and  Briggs  entered  into  partnership  on  Jan.  1,  each  investing 
$8000.  Owing  to  Crown's  greater  experience,  he  receives  2/3  of  the  profits. 
On  July  1,  their  profit  and  loss  account  stands  as  follows. 

Profit  and  Loss. 


Expense $429.60 

Rent 600. 

Discounts 86. 


Merchandise $3129.65 

Discounts 128.46 

Stocks  and  Bonds.       82.50 


Find  the  net  gain  and  present  worth  of  each  partner. 

2.  A  invests  $9000  and  B  his  services,  under  an  agreement  that  the  latter 
shall  receive  $900  yearly  salary,  to  be  paid  from  any  profits,  and  1/4  of  the 
profits  remaining.  The  profits  at  the  end  of  a  year  amount  to  $3126.50. 
Find  each  partner's  present  worth,  B  having  withdrawn  only  one-half  of 
his  salary. 

3.  A,  B  and  C  enter  into  partnership  to  manufacture  advertising 
novelties.  A  invests  $6000,  and  the  other  partners  $4500  each.  The 
partners  agree:  (1)  To  share  losses  equally;  (2)  To  pay  A  $1200  salary 


PARTITIVE  PROPORTION  AND   PARTNERSHIP.    391 

out  of  any  profits,  for  giving  all  his  time  to  managing  the  business;  (3)  To 
divide  any  remaining  profits  in  parts  the  proportion  of  3  (A),  2  (B),  and 
2  (C).  At  the  end  of  the  year  the  profits  are  found  to  be;  on  merchandise, 
$9720;  discounts,  $112.50;  and  the  expenses  of  the  business  exclusive  of  A's 
salary,  $4326.50.     Find  the  present  worth  of  each  partner. 

III.    PROFITS  SHARED  ACCORDING  TO  INVESTMENTS. 
ORAL   EXERCISE. 

1.  A  invests  $5000  and  B  $7000.     What  is  the  partnership  capital? 

2.  Each  partner  invests  what  part  of  the  capital?  Each  should  receive 
what  part  of  any  profits? 

3.  What  should  each  receive  of  $1400  profits?  What  shall  each  pay 
of  $350  loss? 

What  is  each  partner's  share,  according  to  the  following  investments? 

(a)     A,  $6000;  B,  $10,000.       (6)  A,  $5000;  B,  $4000;  C,  $3000. 
Determine  individual  gain,  loss  and  present  worth  in  the  following : 

Investment.  Gain.  Investment.  Loss. 

4.  A,  $2000;  B,  $10,000   $1500.        6.    E,  $3000;  H,  $8000         $3300 

6.    C,  $6000;  D,  $3000      $1860.        7.    G,  $6000;  K,  $1500         $4200 

EXERCISE. 

1.  The  net  profits  of  the  firm  of  Thompson  and  Harris  for  the  last 
year  are  $3584.64.  This  is  to  be  apportioned  to  the  partners'  investments. 
Make  the  apportionments  if  Thompson's  investment  was  $6000  and  that 
of  Harris,  $1700. 

2.  Brown,  Kann  and  Eastman  engaged  in  a  retail  business,  investing 
respectively  $12,000,  $8000  and  $19,000.  Brown,  who  was  the  only 
active  partner,  was  permitted  to  take  $1500  from  the  profits,  annually 
for  his  services,  the  balance  of  the  profits  being  divided  according  to  in- 
vestment. Net  profits  for  the  past  year  were  $7843.12.  Make  the  dis- 
tribution. 

At  the  time  of  apportioning  profits  or  losses  a  business 
statement  is  often  prepared,  showing  a  summary  of  the 
resources  and  liabilities,  the  investments  and  present  worth, 
and,  at  times,  the  losses  and  gains. 


392 


BUSINESS  ARITHMETIC. 


EXERCISE. 
1.    Complete  the  following  business  statement. 
Partnership  Statement. 
Harris  and  Davidson, 


December  31,  19- 


Resources. 
Cash     

5280 

15212 

5000 

8240 

45 
20 

60 

3204 
2500 
? 

Merchandise  on  hand  and  unsold 

Notes  Receivable 

Accounts  Receivable 

Liabilities. 
Accounts  Payable 

50 

Notes  Payable 

Present  worth,  Harris  &  Davidson 

? 

? 

Gains. 
Merchandise 

3640 
1815 
2116 

? 

? 

60 
45 

12760 
521 

?0 

Interest  and  Discount 

60 

Losses. 
Labor 

Rentals 

Expense 

Harris,  Net  Gain        

Davidson,  Net  Gain 

Check. 
Harris,  Investments        

10000 
? 

13000 
681 

— 

? 
? 

Harris,  Net  Gain 

Harris,  Present  Worth 

Davidson  Investment 

Davidson  Withdrawal 

Davidson  Net  Investment 

? 
? 

Davidson  Net  Gain 

Davidson  Present  Worth 

Harris  and  Davidson  Present  Worth 

? 

Note.     Share  profits  according  to  investment. 

^ 

IV.     INTEREST  ON   INVESTMENTS. 

Sometimes  interest  on  investments  is  paid  from  the  profits, 
the  balance  of  profits  being  apportioned  otherwise. 

Illustration.  Example.  A  invests  $6000  and  B  $9000  in  a  business, 
realizing  $3450  profits  the  first  year.  Six  per  cent,  interest  is  allowed  on 
investments,  the  balance  of  profit  being  apportioned  in  the  ratio  of  1  : 2. 
Find  present  worth  of  each  partner. 


PARTITIVE  PROPORTION  AND  PARTNERSHIP-    393 

Solution. 

Interest  on  A's  investment,  1  yr.  on  $6000  =  $360 
Interest  on  B's  investment,  1  yr.  on  $9000  =    540 

900 

$3450  -  $900  =  $2550,  net  profits. 

.  A's  share,  1/3  of  $2550  $    850 

"  B's  share,  2/3  1700 

A's  present  worth,  $6000 +$360 +$850  =         7210 

B's  present  worth,    9000+540  +  1700=     $11240 

EXERCISE. 

1.  Andrews  entered  into  partnership  with  Gates  on  Jan.  1,  1913, 
investing  $6000,  while  Gates  invested  $11,000.  They  agreed  to  allow  each 
other  6%  interest  on  investments,  and  to  share  the  balance  of  profits  in 
the  ratio  of  2  to  3.  Find  the  present  worth  of  each  member  on  Jan.  1, 
1914,  after  the  apportionment  of  profits  of  $6585. 

2.  Jan.  1,  1913,  Carleton  invested  $12,000;  Johnson,  $8500.  Oct.  1, 
1913,  Carleton  withdrew  $3000.  Jan.  1,  1014,  profits  for  past  year 
$4216.50.  Interest  allowed  at  6%,  and  balance  of  profits  shared  equally. 
Apportionment? 

The  average  interest  per  partner  is  sometimes  computed, 
and  each  partner  is  credited  with  the  excess  of  his  investment 
interest  over  the  average,  or  charged  with  any  deficit.  The 
entire  profits  are  then  divided  by  agreement. 

Illustration.     Last  illustrative  example. 
Solution.     Interest  on  A's  investment  $360 
Interest  on  B's  investment     540 
2)900 
Average  interest.  450 

$450  -  $360  =  $90     A's  deficit     (Debit). 
540  -    450  =    90    B's  excess     (Credit). 
A's  1/3  of  profits  =  1/3  of  3450  =  $1150 
B's  2/3  =    2300 

A's  present  worth  =  $6000  +  $1150  -  $90  (deficit)  =    $7,060 
B's  present  worth  =  $9000  +  $2300  +  $90  =  $11,390 

EXERCISE. 

1.    April  1, 1912,  A.  M.  Gates  and  J.  B.  Farquhar  entered  into  partnership 

for  the  purpose  of  carrying  on  a  manufacturing  business.     Gates  invested 

$16,000  and  Farquhar  $12,500.     It  was  agreed  that  interest  should  be 

allowed  and  charged  at  6%,  and  that  gains  and  losses  should  be  divided 


394  BUSINESS  ARITHMETIC. 

equally.  On  Aug.  1, 1912,  Gates  withdrew  $1000,  and  on  Oct.  1,  Farquhar 
withdrew  $500.  On  Apr.  1,  1913,  the  books  were  closed  and  showed  the 
following: 

Gains.  Losses. 

Mdse $15,289  Labor $7240 

Stocks  and  Bonds 986.40        Equipment 1645.20 

Mdse.  discounts 560  Discounts 300 

Expense 1298.75 

Find  the  present  worth  of  each  partner  after  apportionment  of  net  gain. 

2,  Oct.  1,  1912,  M.  Andrews,  B.  T-  Chase,  and  T.  C.  Lewis  formed  a 
partnership.  Lewis  invested  $8500  and  the  other  partners  $7500  each. 
They  agreed  to  allow  and  charge  interest  at  6%  and  to  divide  profits 
equally.  On  Dec.  1,  Lewis  withdrew  $1000  and  Chase  $500.  At  the  close 
of  the  year,  assets  and  liabilities  were  as  follows : 

Assets.  Liabilities. 

Cash $5486.50        Bills  pay $1250 

Mdse 6798.15        Accts.  pay 1670.60 

Real  estate 9725. 

Bills  Rec 3000. 

Accts.  Rec 4520. 

Make  statement  showing  condition  of  business. 

VI.    APPORTIONMENT  BASED   ON   AVERAGE   INVESTMENT. 

The  sum  that  has  the  same  earning  power  for  a  stated 
period  as  two  or  more  sums  invested  for  different  periods,  is 
termed  an  average  investment.  Profits  and  losses  are  often 
apportioned  by  average  investment.  The  apportionment 
may  be  made  by  ratio  or  by  interest. 

Illustration.  Example.  At  organization,  A  invests  $12,000,  but 
withdraws  $2000  at  the  end  qf  8  months;  B  invests  $6000  withdrawing 
$3000  after  6  months.  How  should  the  first  year's  gains  of  $4750  be 
apportioned? 

Solution  (1). 

A's  investment  of  $12,000  for  8  mo.  =  $96,000  for  1  mo. 
A's  investment  of  $10,000  for  4  mo.  =     40,000  for  1  mo. 
A's  average  investment  of  $136,000  for  1  mo. 

B's  investment  of  6,000  for  6  mo.  =     36,000  for  1  mo. 

B's  investment  of  3,000  for  6  mo.  =     18,000  for  1  mo. 

B's  average  investment  =     54,000  for  1  mo. 

A's  average  +  B's  average  =  190,000  for  1  mo. 


PARTITIVE   PROPORTION  AND   PARTNERSHIP.    395 

A's  share  =  136/190  =  68/95. 
B's  share  =  27/95. 

1/95  of  profit  of  $4750  =  $50      68/95  =  68  X  50  or  $3400  =  A's  profit. 

27/95  =  27  X  50  or  1350  =  B's  profit. 
Solution  (2).  Using  any  rate,  say  6%. 
Interest  on  $12,000  for  8  mo.  =  $480 
Interest  on  10,000  for  4  mo.  =  200 
A's  invest,  earning  power  =  $680 
Interest  on  $6000  for  6  mo.  =  $180 
Interest  on    3000  for  6  mo.      =      90 

"$270 
Total  earning  power  $680  +  $270  =  $950,  of  which  A's  mvestment 
represents  68/95  and  B's  27/95. 

Final  step — as  in  previous  solution. 

EXERCISE. 

1.  M.  C.  Barton  and  James  Thompson  form  a  partnership  to  carry  on 
an  agricultural  supply  store.  Barton  invests  $6000  for  9  mo.,  and  then 
adds  $2000.  Thompson  invests  $14,000,  but  withdraws  $4000  at  the  end 
of  4  months.  •  At  the  end  of  a  year  their  summary  accounts  stand  as 
follows: 

Farm  Implements.  Fertilizers. 

Cost  $6240        Sales        $5268.50  Cost  $5692         Sales        $7218.20 

On  hand    3650.  On  hand      965. 

Seeds.  General  Expense. 

Cost  $2132        Sales        $3586.40  Cost  $1296.18 

On  hand    1125. 
Apportion  the  profits  in  accordance  with  average  investment. 

2.  Chas.  Parsons  and  T.  P.  Newton  form  a  partnership  •  to  carry  on 
the  grocery  business.  Complete  the  proprietors'  accounts,  apportioning 
profits  by  average  investment. 

Chas.  Parsons. 
Mar.  1,  Mdse.  $2000  Jan.  1,  Invest  $8000. 

Jan.   1,  Pres.  worth?  Jan.  1,  Net  gain? 

T.  P.  Newton. 
July  1,  Withdraw  $6000  Jan.  1,  1908,  Investment  $18,000 

Jan.  1,  Pres.  worth?  Jan.  1,  1909,  Net  gain  ? 

Profits  and  Losses. 
Losses.  Profits. 

General  Expense $1503.14        Merchandise $9290.68 

Rentals 900  Discounts 587.40 

Labor 1650 

Delivery 2160 

Parson's  net  gain ? 

Newton's  net  gain ? 


396  BUSINESS  ARITHMETIC. 

3,  4,    Solve  the  problems  of  the  last  exercise  by  average  investment. 

GENERAL    EXERCISE. 

1.  Frank  T.  Morris,  R.  P.  Norton  and  Henry  Collins  form  a  partnership 
on  April  1,  to  carry  on  a  furniture  business,  investing  respectively,  $12,000, 
$6000  and  $20,000.  On  June  15,  Morris  withdraws  $500  and  on  Dec.  1, 
Collins  withdraws  $2500.  At  the  close  of  the  first  business  year,  the 
books  showed  the  following  facts:  Cost  of  mdse.  purchased,  $21,690;  sales 
of  mdse,  $32,169.40;  mdse.  on  hand  unsold,  $6,742.10.  Cost  of  real  estate, 
$12,000;  alterations  and  repairs,  $3167.50;  estimated  value  at  close  of 
year,  $14,500;  notes  in  favor  of  firm,  $5640;  accounts  due  the  firm,  $5680; 
accounts  owed  by  the  firm,  $1520;  profits  on  discounts,  $750;  notes  owed 
by  the  firm,  $1200,  mortgage  on  real  estate,  $6000;  general  expense  for 
the  year,  $9650  45;  cash,  $18,811.45. 

Prepare  partnership  statements  at  close  of  year  under  following  con- 
ditions: 

1.  Share  respectively  in  proportions  of  2,  1  and  3. 

2.  Interest  allowed  and  charged  at  5%  and  profits  divided  equally. 

3.  $1200  salary  allowed  Morris,  balance  of  profits  being  divided  ac- 
cording to  original  investment. 

4.  Profits  shared  according  to  average  investment. 

FOR    DISCUSSION. 

1.  Which  system  of  apportionment  is  most  fair,  considering  simply 
investments? 

2.  Which  permits  recognition  of  special  service? 

3.  When  is  it  impossible  to  use  the  method  of  "average  investment"? 


CHAPTER  XLV. 

EQUATION  OF  PAYMENTS  AND  OF  ACCOUNTS. 

INTRODUCTORY    EXERCISE. 

1.  Which  party  to  a  debt  loses  by  its  delayed  payment? 

2.  Who  gains  by  payment  before  due? 

3.  How  may  one  measure,  arithmetically,  this  gain  or  loss? 

4.  $600,  which  I  owe,  is  overdue  1  mo.  Money  being  worth  6%,  what 
ought  I  to  pay  the  creditor  for  the  delay?     Why? 

5.  $200  worth  of  mdse.  bought  Jan.  10,  on  2  mo.  credit,  should  be 
paid  for  on  what  date?  The  creditor  is  entitled  to  how  many  dollars 
extra,  ff  payment  is  made  May  10?  If  paid  Jan.  10  would  the  creditor 
gain  or  lose?     What  might  he  allow  off? 

Debts  draw  interest  from  the  time  they  are  due  until  paid. 
Discounts,  sometimes  equal  to  interest  on  the  debt,  may  be 
allowed  on  payments  before  due.  If  several  payments  are 
due  the  same  party  on  different  dates,  a  single  payment  equal 
to  their  total  may  be  made  on  such  a  date  that  the  overdue 
interest  may  equal  the  discount  on  advance  payments.  This 
date  is  termed  the  average  or  equated  date.  The  process  of 
determining  it  is  termed  averaging  or  equating.  (Why  is  the 
settlement  on  the  equated  date  fair  to  both  parties?) 

EQUATION  OF  PAYMENTS. 
ORAL    EXERCISE. 

1.  The  interest  on  $400  for  6  mo.  =  interest  on  $200  for  ?  mo. 

2.  I  owe  $400  due  in  2  mo.  By  paying  $200  to-day  for  what  extra 
time  might  I  fairly  retain  the  balance? 

3.  I  owe  $600  in  3  mo.,  and  $600  in  5  mo.  When  might  the  total  debt 
be  paid  without  gain  or  loss  to  either  party? 

Illustration  (1).  On  Jan.  10,  A.  C.  Brown  bought  $900  worth  of 
mdse.,  payable,  $200,  Jan.  20;  $400,  Feb.  9;  the  balance,  Mar.  10.     (1)  He 

397 


398  BUSINESS  ARITHMETIC. 

settles  in  full  Mar.  20.  What  does  he  owe?     (2)  On  what  date  would  the 
face  of  debt  be  a  fair  settlement?     Interest  6%. 
Solution  (1).     Debt  paid  Mar.  20. 


Date  Due. 

Payment. 

Overdue. 

Overdue  Interest. 

Jan.  20 

$200 

6'd  da. 

$1.97 

Feb.  9 

400 

39   " 

2.60 

Mar.  10 

300 

10   " 

.50 

Debt    $900 

, 

$5.07 

Int 

.  due          5.07 

Due   $905.07 

Note.     Use  exact  time. 

(2)  By  paying  Mar.  20,  Brown  pays  $5.07  extra.  The  earlier  he  pays 
the  less  interest  he  owes.  The  interest  on  $900  for  1  da.  is  15c.  .".  5.07 
represents  507/15  days  loss  of  interest,  or  34  days.  .'.  The  equated  date 
for  payment  of  $900,  is  34  days  before  Mar.  20,  or  Feb.  14. 

It  is  evident  that  interest  is  found  on  each  payment  from  due 
date  to  date  of  assumed  settlement.  The  total  interest,  divided 
by  the  interest  on  the  total  debt,  gives  the  number  of  days  of 
delayed  payment.  If  there  is  discount  in  place  of  interest,  the 
real  payment  should  be  made  later. 

Illustration.  Find  the  equated  date  for  pajnments  of  $600  due  Mar. 
10;  $300,  April  8;  $600,  May  12. 

Solution.     Assume  a  date  of  settlement,  called  a  focal  date. 

Select  as  focal  date.  May  12,  date  of  last  payment. 

Date.  Payment.  Overdue.  Interest. 

Mar.  10.  $600  63  da.  (to  May  12)       $6.30 

Apr.     8.  300  34  1.70 

May  12  600  0  _0 

Due    $1500  .25)$8.00  due 

Interest  on  $1500  for  1  da.  =  25c.  32,  no.  of 

days. 

.'.  By  paying  May  12,  the  debt  is  overdue  32  days.  32  days  back  from 
May  12  =  Apr.  10,  the  equated  date. 

Check  Solution.  If  Apr.  10  is  correct,  there  should  be  no  gain  or  loss 
by  paying  face  on  that  date.     Take  Apr.  10  as  focal  date. 


Date 

Payment            Overdue        Before  due 

Interest    Discount 

Mar.  10 

$600         (to  Apr.  10)  31  da.      

3.10            

Apr.     8 

300                                2            

.10            

May  12 

600                         32  da. 

3.20 

check 

3.20     =     3.20 

EQUATION  OF  PAYMENTS  AND  OF  ACCOUNTS.    399 

Note.     This  means  that  $1500  may  be  paid  in  full  settlement  on  Apr.  10. 
On  any  date  thereafter  the  amount  due  is  $1500  +  interest  from  Apr.  10. 
Remember:  1.     Any  date  may  be  used  as  focal  date  or  trial  date. 

2.  By  using  first  or  last  date,  one  interest  computation  is  saved.    Why? 

3.  The  face  of  debt  is  always  due  on  the  equated  date. 

EXERCISE. 
Find  the  equated  date,  and  check  by  using  equated  date  or  another 
focal  date. 

1.     A.  B.  Chase,  Dr.  2.     S.  P.  Randall,  Dr. 

1912  1912 

Jan.    16     To  mdse.     $300.  Mar.  12       To  mdse.        $420. 

Jan.   29     To  mdse.       900.  May  16      To  mdse.  500. 

Mar.  12     To  mdse.       400.  Aug.  15      To  mdse.  320. 

What  is  due  Dec.  1? 

3.  James  Field,  Dr. 
1912 

Oct.  16  To  mdse.  $520. 
Oct.  20  To  cash  120. 
Dec.  15  To  mdse.  800. 
1913 

Jan.     3     To  mdse.       320. 

In  the  following,  re-write  the  accounts  by  substituting  the  true  dates 
when  due,  obtained  from  the  credit  terms. 

4.  Robert  Everhart,  Dr.  5.    James  Donnelly,  Dr. 
1912  1912 

Feb.  21     30  da.  credit     $300  Mar.    3     To  mdse.  $240. 

Feb.  25      2  mo.  credit     600  Mar.  18     To  mdse.,  30  da.    380. 

Mar.  16       1  mo.  credit     900  Mar.  30     To  mdse.,  2  mo.     456.40 

Apr.     4     To  mdse.,  60  da.    820.60 
Apr.  16    To  mdse.  900. 

6.     Jones  &  Co.,  Dr. 
■  1912 

Apr.  16     Mdse.     60  da.     $456.10 

Apr.  29     Mdse.     30  da.       829.30 

May    5     Mdse.      3  mo.     420.00 

June  16     Mdse.    36.50 

EQUATION  OF  ACCOUNTS. 
Accounts  having  debits  and  credits  are  equated  to  determine 
the  due  date  of  the  balance.     The  most  convenient  focal  date 
is  the  latest  named  in  the  account. 


400 


BUSINESS  ARITHMETIC. 


Illustration. 


James  C.  Westcott. 


1909 

Mar. 

12 

Mdse. 

$360 

Mar. 

30 

By  cash 

$300 

29 

300 

Apr. 

10 

By  note 

120 

May 

4 

240 

(Balance 

$480 

Solution.  Here  we  have  both  debit  and  credit  interest,  the  former 
being  charged  on  debts  overdue,  and  the  latter  being  credited  on  pay- 
ments made  to  focal  date. 


Focal  date.  May  4. 

Date. 

Debit  Amounts. 

Period  (to  May  4). 

Dr 

.  Interest. 

Mar.  16 

$360 

49  da. 

$2.94 

Mar.  29 

300 

36  da. 

1.80 

May    4 

240 

$900 

$4.74 

Credit  Amounts. 

Cr 

.  Interest. 

Mar.  30 

$300 

35  da. 

$1.75 

Apr.  10 

120 

24  da. 

.48 

$2.23 


$420 

Balance  =  $900  -  $420  =  $480. 

Net  interest  =  Dr.   int.  -  cr.   int.  =  $4.74  -  $2.23  =  $2.51,  dr.  int. 
Since  the  interest  is  debit,  payment  is  overdue. 
Interest  on  $480  for  1  day  =  8c. 

2.51  -^  .08  =  31  +,  the  number  of  days  overdue. 

May  4  —  31  days  =  Apr.  3,  equated  date. 

Check.    Assume  Apr.  3  as  focal  date. 


Date. 

Amount. 

Period. 

Dr.  Int. 

Cr.  Int. 

Mar.  16 

$360 

18  da. 

$1.08 

Mar.  29 

300 

5  da. 

.25 

May    4 

240 

31  da.  (before  due) 

$1.24 

Mar.  30 

300 

4  da.  (before  due) 

.20 

Apr.  10 

120 

7  da.  (delayed) 

1.14 

$1.47 

check 

$1.44 

Note.     The  difference  between  dr.  and  cr.  interest  =  the  remainder 
in  first  solution,  when  dividing  total  interest  by  interest  for  one  day. 


EXERCISE. 

Find  and  check  the  equated  date. 

1.  C.  P.  Newton. 


1912 

1912 

Mar.  16     To  mdse. 

$1400 

Apr.  4 

May  28 

2200 

June  6 

By  cash 


$  200 
1000 


EQUATION  OF  PAYMENTS  AND  OF  ACCOUNTS.    401 


2. 


The  Newbold  Co. 


1912 

Nov.  19     To  mdse.  $26,580 

30  "  41,590 

f)ec.  12  "  16,250 

30  "  38,845 
1913 

Jan.   20  "  268 


1912 
Nov. 
Dec. 
Dec. 


30 
15 
31 


By  cash 


Note 


$200 
300 
500 


3. 

Chas.  p. 

Randall. 

1912 

1912 

July  15 

To  mdse.,  30  da.      $360 

Aug.    5 

By  cash 

$400 

31 

15  da.        180 

Sept.  10 

Note 

300 

Aug.  12 

30  da.        420 

10 

Cash 

300 

Sept.    6 

2  mo.       620 

4. 

James  Brownet.t.  &  Bro. 

1912 

1912 

Feb.  28 

By  note 

$  300 

Feb.  16    By  mdse.  60  da.  $325 

Mar.  16 

Mdse. 

120 

20          "          30  da.   244.60 

Apr.     2 

Cash 

400 

Mar.  18          "         3  mo.    840 

May  10 

(( 

600 

29          "         2  mo.    628 

Note. 
accounts. 


Account  sales  are  often  averaged  m  the  same  manner  as  general 

Washington,  D.  C,  May  16,  1913. 
Account  Sales  op  Potatoes. 

Sold  for  account  of  James  Parker, 

Hyattsville,  Md. 
By  Robert  Harper, 

Commission  Merchant. 


1913 

Sales. 

Apr.  30 

20  bbl.  potatoes. 

$2.40 

May    6 

50    "           " 

3.00 

8 

120    " 

2.60 

12 

10    " 

Charges. 

2.00 

Apr.  28 

Freight  and  cartage 

.46 

May  12 

Storage 

.30 

12 

Commission,  5%  of  sales 

? 

? 

Net  proceeds,  due  by  equation 

? 

27 

402  BUSINESS  ARITHMETIC. 

CASH  BALANCE. 

The  cash  balance  of  an  account  on  any  particular  date  is  the 
amount  due  on  that  date.  If  interest  and  discount  are  not 
allowed,  the  cash  balance  is  the  same  as  the  account  balance; 
if  allowed,  it  equals  the  regular  balance  plus  or  minus  the  net 
interest  or  discount;  or  it  equals  the  balance  of  the  account 
plus  interest  for  the  time  between  equated  date  and  average 
date,  if  later  than  equated  date;  or  minus  interest,  if  given  date 
is  earlier.  The  first  illustrative  example,  page  398,  shows  the 
cash  balance  on  March  20. 

EXERCISE. 

Find  the  cash  balance  directly  for  each  of  the  accounts  in  the  preceding 
exercise,  checking  by  using  equated  date. 
Ac.  1.     What  is  due  July  1? 

2.  What  is  due  Jan.  8? 

3.  What  is  due  Oct.  31? 

4.  What  is  due  Jan.  1,  1913? 
6.    What  is  due.July  1? 


CHAPTER  XLVI. 


BILLING. 


INTRODUCTORY    EXERCISE. 

1.  Bring  to  class  several  bills  for  goods  purchased.  State  the  infor- 
mation that  each  heading  gives. 

2.  What  information  is  given  concerning  the  articles  bought? 

3.  Who  makes  out  a  bill  for  goods?  What  becomes  of  it?  Trace 
its  life. 

4.  Of  what  value  is  the  bill  to  the  purchaser?  What  might  happen  in 
case  no  bill  was  given? 

The  hill  of  merchandise  is  the  statement  in  detail  of  mer- 
chandise sold  in  the  course  of  trade,  and  is  prepared  by  the 
seller  for  the  buyer.  Formerly  the  term  invoice  was  applied 
to  statements  of  merchandise  sold  wholesale,  and  of  incoming 
goods  from  abroad.  Now  the  terms  are  frequently  used 
interchangeably. 

I.    SIMPLE  BILL. 

Washington,  January  11,  1913. 
C.  M.  Brown, 
Groceries  and  Provisions. 
Sold  to  Henry  Eastman, 

1426  13th  Street. 


1 

1 

sk.  Flour 

85 

2 

6 

lb.  Lard 

@.17 

1 

02 

3 

U 

pk.  Apples 

.72 

4 

10 

b.  Gran.  Sugar. 

.06^ 

5 

2 

qt.  Molasses 
lb.  Rice 

.18 

6 

3 

.09 

7 

2 

lb.  Starch 

.05 

8 

Yeast 

02 

9 

? 

10 

11 

Paid, 

12 

C.  M.  Brown 

13 

14 

403 


404  BUSINESS  ARITHMETIC. 

EXERCISE. 

1.  State  in  your  own  words  just  what  took  place  according  to  the 
bill  on  page  403. 

2.  Are  there  any  non-essentials  in  this  bill?    Add  two. 

3.  If  settlement  is  not  made,  how  is  the  bill  affected?    How,  if  sent 
C.  O.  D.? 

4.  What  is  the  advantage  of  numbering  the  items  on  the  bill? 

5.  Rule  up  three  bill  forms  and  prepare  threo  original  bills  for  a  Satur- 
day's marketing.     Secure  reliable  information  as  to  quantities  and  prices. 

Receipting  for  Payment. 
When  payment  is  made,  bills  are  receipted  by  the  creditor 
or  by  the  party  representing  him,  who  writes  the  word  "  Paid," 
or  some  equivalent  expression,  across  the  face  of  the  bill  over 
his  signature. 

EXERCISE. 

1.  Why  is  the  creditor's  signature  necessary? 

2.  Examine  the  forms  of  receipt  of  the  model  bills  in  this  section,  and 
throughout  the  book,  and  determine,  in  each  case: 

(a)  What  facts  are  stated  in  the  receipt. 

(6)  By  whom  receipted  and  under  what  probable  conditions. 

(c)  What  part  of  the  receipt  may  be  printed. 

(d)  Under  which  of  the  following  classes  the  receipt  belongs: 

(1)  Receipt  by  the  direct  creditor. 

(2)  Receipt  in  the  firm  name  by  a  partner. 

(3)  Receipt  of  a  corporation  by  an  officer. 

(4)  Receipt  for  the  direct  creditor  by  a  third  party. 


BILLING. 


405 


11.    SIMPLE  TWO  COLUMN  BILL. 

January  31,  191 

Chestnut  Grove  Diary, 

1421  Oak  Street. 

John  C.  Norton,  Jr. 

Sanitary  Cream  and  Milk. 

Sold  to  James  Fielding 

646  Pine  Street. 

Acc't  Rendered 

132  Qts.  Milk                            .09 
6  Pts.     "                                .05 
Pts.  Gilt-Edge  Cream 
7^  Pts.        "              "             .19 
Gills       " 
Pts.  Choice           " 
Hf.  Pts.  Choice     " 
Gills     " 

1  A    Z^+o,         'Dii4^4^y>.««^i1U                               AO 

Received  payment 

John  C.  Norton,  Jr. 

EXERCISE. 

1.  Extend  the  above  bill.  How  should  the  form  be  changed  if  $3.00 
is  paid  on  account?     How,  if  nothing  is  paid? 

2.  You  are  directed  by  James  Kent  to  deliver  daily  2  qt.  1  pt.  of  milk 
and  1  pt.  of  cream.  In  addition,  you  deliver:  Jan.  5,  3  qt.  milk;  Jan.  8, 
2  qt.  milk;  1  pt.  cream;  Jan.  21,  2  qt.  1  pt.  milk;  1  pt.  cream.  Prices:  9c 
per  qt.  for  milk;  16c  per  pint  for  cream.  Render  a  bill  and  have  your  clerk 
receipt  for  payment. 

3.  Design  an  original  bill  for  similar  use  for  an  ice  company. 

4.  In  what  other  businesses  could  similar  bills  be  used? 

Terms  of  Sale. 

As  is  evident  from  the  problems  and  illustrations  given,  the 
terms  of  payment  on  bills  of  goods  may  vary  markedly  with 
the  character  and  quantity  of  sales.  The  list  of  terms  that 
follow  are  selected  from  actual  bills. 


Cash. 
Net  cash. 


List  of  Terms  op  Sale. 


406  BUSINESS  ARITHMETIC. 

3.  Cash  or  exchange  on  New  York. 

4.  Account. 

5.  Interest  after  30  days. 

6.  30  days;  2/10. 

7.  Net  30  days;  note  to  your  own  order,  payable  at  a  Philadelphia 

bank. 

8.  Net  90  days;  5%  discount  for  exchange  on  New  York  in  30  days. 

9.  30  days.     If  paid  in  10  days,  2%  discount. 

10.  30  days. 

11.  30  days.    If  not  promptly  paid  interest  will  be  charged  from  date  of 

sale. 

12.  Cash,  5%;  30  days,  2%;  60  days,  net. 

13.  Ten  day  note. 

14.  60  days  from  Aug.  10;  2%  10  days. 

15.  5%  30  days,  or  6%  10  days.     Payable  in  New  York  Gty  funds. 

16.  Due,  October  2.     Less  2%  if  paid  in  10  days. 

17.  C.  O.  D. 

18.  5/60;  6/30;  7/10. 

19.  Best  discount  allowed  for  unexpired  time  is  at  the  rate  of  6%. 

20.  Net  30  days,  or  1%  discount  in  30  days,  or  2%  discount  in  10  days. 

New  York  funds. 

21.  Net  after  60  days. 

22.  Subject  to  sight  draft  after  60  days.    Payable  in  gold  or  its 

equivalent. 

23.  Net  30  days;  2%  discount  10  days.     No  discount  allowed  after 

,  191—. 

24.  Accounts  due  the  first  of  the  month. 

25.  2%  discount  if  paid ,  191—;  Due  net, ,  191—. 

EXERCISE. 

1.  Give  your  understanding  of  the  meaning  of  each  of  the  terms  stated 
above. 

2.  Examine  any  bills  that  you  can  secure  and  see  whether  you  can  add 
to  the  above  list. 

3.  State  the  effect  of  each  set  of  terms  on  the  settlement  of  a  bill  of 
merchandise  amounting  to  $1600,  purchased  August  1. 


BILLING. 


407 


III.    FORM  OF  MANUFACTURER'S  INVOICE  AND  LETTER  OP 
TRANSMISSION. 


Address    Albany,  N.  Y. 


Invoice. 
Name  John  Smithson 
Order  No.  17296 

Bought  of  WASHINGTON  FLOUR  CO. 
Manufacturers 
Fancy  Patent  Flours 
Buffalo,  N.  Y. 
Ship  to  Order  Washington  Flour  Co.  Date  12/30/191- 

Destination  Albany,  N.  Y. 
Via.  N.  Y.  C.  R.  R. 


Bbl8. 

H. 

Bbls. 

No.Sks. 

Sz.Sks. 

Kind. 

Brand. 

Marks. 

Pricfi 
PerCwt. 

Amount. 

180 

50 

200 
80 
40 

260 

24 

49 

98 

140 

Paper 
Cott'n 
Grain 
Jute 

Ames 

Buffalo  Prime 
Conrad 
Golden  Queen 
Harvest 
Bakers  Pride 

5.65 
4.90 
4.60 
4.80 
4.40 
4.45 

? 
? 
? 
? 
? 
? 

? 

Car 
47329 

Initials 
L  V.  R.  R. 

Date 
12/30/191— 

The  above  is  an  invoice  of  goods  shipped  this  date,  in  accordance  with 
your  order  of  12/17/191— 

Washington  Flour  Co. 

per  Temple. 
EXERCISE. 

1.  Extend  the  bill  and  compare  with  previous  bill,  as  to  columns,  entry 
of  items,  etc. 

2.  Explain  the  advantages  and  disadvantages  of  the  use  of  many 
columns. 

3.  From  the  standpoint  of  the  customer,  what  portion  of  the  heading 
might  be  dispensed  with? 

4.  Prepare  a  similar  original  invoice  for  a  wholesale  order,  using  news- 
paper market  quotations.  Add  terms.  Receipt  the  bill,  signing  your 
name  as  officer  of  the  selUng  company. 


408 


BUSINESS  ARITHMETIC. 


IV.    CLOTH  MANUFACTURER'S  INVOICE. 

Goods  Purchased  will  not  be  Taken  Back  Except  for  Damage  or  Imperfection. 
No  Claims  allowed  unless  reported  within  10  days  after  receipt  of  goods. 

MORRIS    &    NORTON 

DRESS  GOODS 

New  York,  May  21,  191— 
John  Harris  &  Co.  Terms 

Register  1337  N.  Y.  City.  S  per  cent.  30  das.,  or 

Book  45     Folio  14  6  per  cent.  10  das. 

THIS  INVOICE  PAYABLE  IN  NEW  YORK  niTY  FUNDS 


19  Pes  NORI^OLK  LT  PERCALE 
45-i-  45^  45^  45^  45^  45^  46^ 
46^  45^  46^  46^  45^  45^  45^ 
46^  45-^  46^  46^  45^ 

16  Pes  E-QREOLK  PK  PERCALE 
45   45-3^  45^  46^  45-^  45^  45^ 
45^  46^  45-^  46^  45^  45^  46^ 
45-^  45^ 

17  Pes  -\^  PEPPERELL  BROWN 
45  45^  46^  45-^  45^  45-^  45^ 
45^  45-i-  45^  46^  45^  45^  45^ 
45^  45-^  45^ 


10* 


10 


09 


EXERCISE. 

1.  Extend  the  bill.  What  amount  will  settle  it  June  19?  June  24? 
August  23?     October  16?    On  what  date  is  the  face  value  due? 

2.  On  October  19,  the  Fall  River  Manufacturing  Co.  sells  you,  on 
terms  of  4/60;  5/30;  6/10;  36  pc.  C.  K.  Checks,  45,  46,  47S  48,  45^,  46S 
42',  412,  452^  46^  44^  453^  442^  46^  47^  431,  45',  42^,  43^,  44S  42',  46,  47,  48«, 
45,  41',  42,  44,  47',  45^,  43^,  44S  45^,  43',  42',  44  @  23c. 

Also  12  pc.  Ginghams:  45S  47,  46^,  43,  45,  49^,  50,  48',  47,  49,  51»,  49« 
@9f. 

Draw  up  the  bill  and  receipt  for  the  company  for  settlement 
on  November  16. 


BILLING. 


409 


3.  Ascertain  by  inquiry  the  names,  common  piece  lengths  and  prices 
of  four  standard  cloths  and  draw  up  an  original  bill.  Name  your  own 
terms  and  receipt  for  payment. 

V.    DISCOUNT  BILLS. 
See  form  on  page  242. 
Claims  for  errors  and  over- 
charges   must    be    made 
within  ten  days  of  delivery. 
Terms,  30  days  net;  5%  cash.  Louisville,  October  25, 191- 

THE  KENAWHA  SUPPLY  COMPANY. 

GENERAL   CONSTRUCTION   AND  RAILWAY   SUPPLIES. 

1468  River  Street. 


Your  Order  No.  6657 

Your  Req.  No 

Shipped  L  L.  Frt.  10/25/1- 
Our  No.  'D-S456 
Salesman  Roberts. 


Sold  to  John  P.  Carey, 

Evanston,  IlL 


Quan- 
tity 
Fur- 
nished. 

Description. 

List. 

Disc,  i 

Extension. 

Total. 

16 

sq.  10X14  Gal.  Sheet  Roofing 
ft.  Gal.  Valley 
ft.  Ga .  Climax 

lb.  W.  P.  R.  Wire 

C.  Machine  Bolts,  \  X3^, 
C.  Machine  Bolts,  5/8X4 
C.  Machine  Bolts,  fX8 

6 

Le 

4 

7 

13 

87 
11 
14 

ss 
12 

72 
15 
85 

20% 
30% 

50,  10 

5% 

164 

48§ 

IfiO 

2 

1^ 

Trade 

1 

Received  payment,  October  17,  191 — 

THE  KENAWHA  SUPPLY  CO. 

J.  Towne,  Treas. 
Extend  and  explain  bill. 

How  does  it  differ  in  arrangement  from  the  bill  on  page  242? 

Prepare  discount  bills  for  examples  on  pages  237,  239  and  242. 


Supply  names  of  customers  and  dealers  where  necessary. 


410 


BUSINESS  ARITHMETIC. 


VI.     MONTHLY  STATEMENT. 

Retail  Statement. 


Samuel  Henderson, 

The  Morrison. 


Chicago,  111.,   Sept,    30,   Wl- 


IN   ACCOUNT  WITH 

HENRY  NORTON  &  COMPANY. 


Items. 

Chargea. 

Credits. 

Sept. 
1 
2 

To  Balance  (Bill  rendered) 
12  yd.  Lawn         .18 

8  Lawn  Mull    .30 

3  iDt.  Ribbon      1.15 
6  yd.  Embroidery   .28 
2  yd.  Elastic      .12 
5     Binding      .15 

4  yd.  Linen        .65 
1  set  Purs, 

1     Ermine  Cape 
12  yd.  Silk       1.25 
1     Set  Furs 
1  rem. Gingham 

9  yd.  Calico       .10 

5  C.  Thread    .10 

1  rem. Gingham 

2  Shirtwaists 

3  pr.  Blankets    4.00 
2  pr.  •  Pillows     2.25 
2     Bolsters    2.50 

— 

6Q 

40 

165 

5 



6 

— 

12 

165 
240 

00 
00 

*Cr. 
24 

65 

00 

Cr. 
27 

9 

60 

65 

Le 

ss 



— 

*NoTE  1.     Credit  items  are  usually  entered  in  red. 

Note  2.  Where  individual  purchases  have  covered  many  items  on 
any  one  date,  and  a  bill  has  been  rendered  at  the  time,  details  are  not 
entered  on  the  statement,  but  the  date  of  each  purchase,  the  words  "to 
Bill  Rendered"  and  the  amount  of  individual  bill  are  entered.  This  ia 
common  in  wholesale  statements. 


BILLING. 


411 


EXERCISE. 

1.  How  does  this  statement  differ  from  an  ordinary  bill  of  goods? 

2.  Extend  the  statement. 

3.  Explain  just  what  has  taken  place  according  to  the  statement 
How  does  it  happen  that  credits  occur? 

4.  Should  the  statement  be  receipted  in  case  of  partial  settlement? 

5.  Prepare  an  original  statement  of  a  month's  purchase  of  groceries. 
Receipt  for  payment. 

EXERCISE. 

1.     Draw  up  a  " to  bill  rendered  "  statement  from  the  following  account: 

Robert  H.  Lawrence, 

1356  18th  Street,  N.  W. 


191— 

— 

191— 

1 

Mar.  3 

Mdse. 

1     42 

16 

Mar.  8 

Cash 

12  00 

18 

tt 

121 

15 

21 

Mdse.  ret. 

540 

23 

Debit  memo. 

4 

19 

23 

Overcharge 

48 

27 

Mdse. 

16 

35 

30 

10  da.  dft. 

150  00 

28 

a 

39 

12 

31 

Cash 

50  00 

30 

(I 

125 

04 

31 

11 

11 

85 

2.  Draw  up  an  original  statement  to  cover  one  quarter  of  a  year, 
containing  at  least  fifteen  debit  items  and  six  credit  items  of  different 
character. 


VIII.  REQUISITION. 
The  requisition  is  a  combined  order  and  voucher  or  bill, 
used  in  ordering  supplies  for  offices  and  workshops,  when  the 
approval  of  a  higher  official  is  necessary.  Sometimes  it  does 
not  involve  prices,  but  is  simply  a  request  from  an  official  or 
foreman,  drawn  on  the  stock  clerk,  for  supplies  needed,  which 
the  latter  keeps  in  stock  ready  for  use  as  required.  In  the 
illustration  (page  412),  however,  the  articles  required  must 
be  purchased.  The  superintendent  of  a  certain  department 
asks  for  the  supplies,  and  the  general  manager  approves  the 
request  and  directs  the  purchase.  Requisitions  vary  greatly 
in  different  businesses  and  government  offices. 


412 


BUSINESS  ARITHMETIC. 


Requisition  Book. 

March  10, 191— 

THE  RANKIN  MANUFACTURING  COMPANY. 

Charge  to  the    Repair    Shop;  Model  Department. 

Please  furnish  for  use  in  this  shop  the  materials  specified  below: 

Henry  C.  Carter, 
Superintendent. 


Items. 


Quantity. 


400  b.  f. 

2000  b.  f. 

600  b.  f. 

5qt. 

9qt. 

2  1b. 

101b. 

181b. 


Material. 


W.Pine,  2X3-6', 
Oak,  1X9,  per  M. 
Chestnut,  per  M. 
M.  Y.  Varnish 
H.  O.  Finish 
Indian  Red 
Vermilion  Red 
Yellow  Murdock 


perM. 


$42.00 
50.00 
36.00 
.21 
.24 
.10 
.24 
.26 


Estimated 
Cost. 


March  11,  191— 
Approved  and  ordered  purchased. 

Jas.  M.  Dean, 
Gen'l  Man. 


Received  the  above  mentioned 
material  on  March  16,  191 — 

Henry  C  Carter, 
Supenntendent. 


EXERCISE. 

1.  Extend  the  form. 

2.  State  exactly  what  has  taken  place. 

3.  Prepare  an  original  requisition  for  furniture  and  material  required 
for  the  equipment  of  some  business  office.  Sign  as  subordinate  and  have 
some  other  pupil  sign  as  a  superior  officer. 


FOR  INVESTIGATION  AND  REPORT. 

EXERCISE. 

1.  Collect  specimen  bills  and  statements  from  at  least  thirty  distinct 
businesses.  Classify  these  {a)  in  order  of  complexity;  (6)  according  to 
the  difficulty  of  the  arithmetical  computations  involved. 

2.  Write  a  brief  on  the  information  furnished  by  the  bill  headings. 
Compare  headings  of  the  bills  of  retail,  wholesale  and  manufacturing  houses. 

3.  From  your  study  of  the  bills  discover  how  the  "terms  of  payment" 
vary  with  the  character  of  the  business. 


BILLING.  413 

4.  What  knowledge  of  arithmetic  should  a  bilhng  clerk  have? 

5.  Study  and  report  on  the  use  and  value  of  special  columns.     Give 
illustrations. 

6.  Collect,  and  submit  an  illustrated   report  on  forms  which,  while 
not  bills,  are  worked  out  on  the  same  principle. 

7.  Show  how  each  bill  form  is  designed  to  meet  the  needs  of  the  business 
in  which  it  is  used. 

8.  Examine  bills  and  see  how  many  different  ways  of  entering  items 
you  can  discover. 

9.  Report  on  the  different  forms  of  receipt  for  payment  that  you  can 
discover.     Explain  the  exact  meaning  of  each 


CHAPTER  XLVII. 

STORAGE. 

INTRODUCTORY    EXERCISE. 

1.  What  circumstances  may  lead  a  housekeeper  to  store  her  fumitiu'e? 

2.  Why  does  a  commission  merchant  place  certain  merchandise  in 
"cold  storage"?     Why  does  he  not  own  a  storage  plant? 

3.  Why  is  grain  "stored"  in  grain  elevators? 

4.  If  I  rent  a  room  in  a  furniture  warehouse  from  July  15  to  October  25, 
at  $9.00  per  30  day  month,  or  fraction  thereof,  the  storage  bill  is  $ . 

5.  The  cold  storage  of  40  cases  of  eggs  for  50  days,  at  40c  per  month, 
or  fraction,  costs  what  amount? 

Storage  is  the  charge  for  storing  goods  in  an  elevator  or 
warehouse.  It  is  computed  on  quantity  rather  than  on  value. 
The  term  of  storage  is  the  time  for  which  the  storage  rate  is 
quoted.  The  rate  may  be  for  the  day,  week,  month,  or  period 
of  30  days,  etc.  Fractions  of  terms  are  usually  considered  as 
full  terms. 

Storage  warehouses  and  grain  elevators  are  closely  regu- 
lated by  law.  Often  rates  are  fixed  by  law,  by  Chambers 
of  Commerce,  associations  of  warehousemen,  etc.  In  the 
storage  of  grain,  and  of  certain  other  produce,  there  are  strict 
regulations  for  grading  receipts  of  merchandise,  since  the 
person  storing  cannot  expect  to  withdraw  the  identical 
property  he  deposited. 

SIMPLE  STORAGE. 
Simple  storage  is  storage  computed  at  the  time  of  withdrawal 
of  goods. 

EXERCISE. 

(Solve  mentally  if  possible.) 
Compute  the  storage  on: 

414 


STORAGE. 


415 


1.  200  cs.  eggs,  for  3  mo.  at  10c  per  mo. 

2.  500  cs.  eggs,  for  4  mo.,  at  10c  per  mo.  for  2  mo.,  and  8c  per  mo. 
for  succeeding  months. 

3.  800  lb.  of  cheese  from  August  17  to  October  26,  at  8c  per  100  lb. 
per  month  of  30  days. 

4.  The  storage  rate  on  the  following  bill  is  9c  per  100  lb.  per  30  da. 
Extend  the  bill,  explaining  each  item. 

March  31,  191— 
Messrs.  James  Brown  &  Son, 

To  The  Morton  Storage  Co.,  Dr. 


Article. 

Quantity. 

Quantity. 

Time. 

Kate. 

Lot 
No. 

In. 

Out. 

i 

Amt. 

6746 

Cheese 

20000  lbs. 

Jan. 

16 

Feb. 
Feb. 
Mar. 
Mar. » 

15 

28 

6 

28 

5000  lbs. 
8000 
2000 
5000 

30 

43 

49 

? 

1 

2 
2 
? 

9c 
18c 

? 
? 

4.50 
? 
? 
? 

? 

5.  Prepare  a  bill  for  the  storage  of  160  cs.  eggs  on  Jan.  15,  by  James 
Quick,  at  the  rate  of  10c  per  month  of  30  days.  Deliveries:  Jan.  28,  40  cs.; 
Feb.  19,  50  cs.;  the  balance.  Mar.  13. 

6.  Compute  the  storage  on  5000  lb.  poultry,  stored  Mar.  15  at  l/4c 
per  lb.  per  30  da.  DeUveries:  Mar.  30,  1500  lb.;  Apr.  4,  800  lb.;  May  12, 
2000  lb.;  June  16,  the  balance. 

7.  Extend  this  bill.     Rates:  10c  per  bale,  15c  per  cs.,  per  30  da. 

October  31,  19— 
Messrs.  Cates  &  Co., 

To  The  Commercial  Storage  Co.,  Dr. 


Quan- 
tity. 

Marks  and  Nos. 

Rec'd. 

Deliv. 

Bate. 

Amount. 

8 

4 

12 

6 

bales  No.  675-83  A 
Cs.  c  D  No.  721-24 
bales  A.  C.  O  No.  829-41 
bales  BC  <S>  No.  726-31 

June 
July 

8 
16 
27 

5 

Oct. 
Aug. 
Sept. 
Oct. 

20 
19 
30 

4 

? 
? 
? 
? 

In  cases  where  goods  are  being  constantly  received  and 
delivered,  and  simple  storage  is  charged,  it  is  assumed  that  all 
deliveries  are  made  from  goods  longest  in  storage. 


416  BUSINESS  ARITHMETIC. 

Illustration.     Robert   Connor  stores  500  bbl.   Apr.   16;  300  bbl. 
May  1;  200  bbl.  May  19.     He  withdraws  400  bbl.,  Apr.  30;  200  bbl. 
May  18;  400  bbl.,  May  27.     Compute  his  storage  at  the  rate  of  6c  per 
month,  or  fraction. 
Solution. 
Date        Rec'd        Deliv.  Rate. 

Apr.  16     500  bbl. 

30  400  bbl.        Stored,         Apr.  16-30,  14  da.  6c 


May    1     300 


1R  rtrjn  f  100  Stored,  Apr.  16-May  18,    32  da.    12c 

^°  "^^^  \  100  Stored,  May    1-18,             17  da.     6c 

19    200 

97  .^n  J  200  Stored,  May    1-27,             26  da.     6c 

—-  ^^  1200  Stored,  May  19-27,               8  da.     6c 

1000  1000  Total  storage, 


Cost. 
S24.00 


12.00 
6.00 

12.00 
12.00 


$66.00 


EXERCISE. 

1.  A.  C.  Bronson's  storage  memorandum  is  as  follows:  Stored,  Nov.  12, 
2000  lb.  poultry;  Dec.  5,  2000  lb.;  Jan.  18,  400  lb.  Withdrew,  Dec.  24, 
2200  lb.;  Jan.  10,  1500  lb.;  Mar.  17,  700  lb.  Compute  the  storage  at  the 
rate  of  l/4c  per  lb.  per  month. 

2.  John  T.  Bartlett's  memorandum  is:  Stored,  Aug.  18,  50  bbl. 
potatoes;  Sept.  15,  200  bbl.;  Oct.  17,  300  bbl.;  Withdrawn,  Nov.  5,  200 
bbl.;  Nov.  20,  100  bbl.;  Dec.  5,  100  bbl.;  Dec.  20,  the  balance.  Compute 
his  storage  bill  at  the  rate  of  6c  per  bbl.  per  month. 

3.  On  Oct.  5,  Brown  stored  100  bbl.  apples;  Oct.  20,  50  bbl.;  Nov.  16, 
180  bbl.  Nov.  1  he  withdrew  80  bbl.;  Dec.  1,  50  bbl.;  Jan.  5,  100  bbl.; 
Feb.  9,  the  balance.  Compute  the  storage  at  the  rate  of  12c  per  bbl.  per 
month. 

2.    AVERAGE  STORAGE. 
When  receipts  and  deliveries  are  frequent,  it  is  a  custom  to 
average  the  time,  and  to   charge  average  storage.    Usually 
exact  time,  with  30  day  periods,  is  used. 

1.  The  storage  of  200  bbl.  for  one  month  equals  the  storage  of  1  bbl. 
for  ?  months. 

2.  The  storage  of  100  bbl.  for  2  months  equals  the  storage  of  1  bbl.  for 
?  months. 

3.  The  storage  of  500  bu.  for  20  days  equals  the  storage  of  1  bu.  for 
?  days. 

Illustrative  Example.  The  following  is  a  memorandum  of  flour 
stored  by  C.  P.  Dean  with  the  Commercial  Storage  Co.,  at  a  rate  of  4c 


STORAGE.  417 

average  storage.  Receipts:  Feb.  6,  200  bbl.;  Feb.  21,  150  bbl.;  Mar.  8, 
400  bbl.;  Mar.  29,  200  bbl.  DeUveries:  Feb.  12,  100  bbl.;  Mar.  9,  150  bbl.; 
Mar.  21,  300  bbl.;  Apr.  4,  400  bbl. 

Solution.    Arrange  entries  in  order  of  dates : 
Date.        Rec'd.        Deliv.  Balance        Time.  Eqiiiv.  to  1  bbl. 

Stored  for 
Feb.     6    200  bbl.  200  bbl.        6  da.  1200  da. 

12  100  bbl.       100  9  900 

21     150  250  15  3750 

Mar.    8    400  650  1  650 

9  150  500  12  6000 

21  300  200  8  1600 

29    200  400  6  2400 

Apr.     4  400  0 

The  storage  items  are  equivalent  to  the  storage  of  one 

barrel  for 16,500  days. 

16,500  days  =  550  terms  of  30  days  each 
At  4c  per  term,  the  storage  =  550  X  4c  =  $22.00.    Ans. 
Note.     Notice  that  each  "one  bbl."  equivalent  is  the  product  of  the 
balance  on  hand  by  the  number  of  days  that  balance  remains  unchanged. 

EXERCISE. 

1.  Solve  the  examples  in  the  last  written  exercises  by  average  storage. 

2.  Are  charges  lower  by  average,  or  by  simple  storage?    Why? 

3.  Compute  average  storage  at  the  rate  of  l^c  per  bu.  per  30  days, 
from  the  following  memorandum:  Received:  Oct.  10,  12,000  bu.;  Oct.  25, 
14,000  bu.;  Nov.  1,  3500  bu.  Deliveries:  Dec.  5,  2500  bu.;  Dec.  8, 3000 bu.; 
Dec.  21,  5,000  bu.;  Dec.  30,  8,000  bu.;  Jan.  5,  11,000  bu. 

4.  On  Jan.  16,  Robert  Osbom  bought  2,000  bbl.  of  flour  at  $4.25  and 
placed  it  in  storage  at  an  average  rate  of  5^c  per  barrel.  On  Jan.  29, 
he  bought  and  stored  200  bbl.  at  $4.30;  on  Feb.  16,  he  withdrew  300  bbl. 
for  sale  at  $4.65;  on  Mar.  11,  500  bbl.  for  sale  at  $4.80,  and  disposed  of  the 
balance,  on  Apr.  3,  at  $5.10.  Drayage  charges  amounted  to  15c  per  bbl. 
What  was  his  per  cent,  of  profit? 

FOR   INDIVIDUAL   REPORT. 

1.  Prepare  a  brief  on  the  knowledge  of  arithmetic  required  by  the 
average  retail  clerk. 

2.  Prepare  a  brief  on  the  knowledge  of  arithmetic  required  by  the 
average  mechanic. 

28 


418  BUSINESS  ARITHMETIC. 

3.  Report  on  "What  Knowledge  of  Arithmetic  is  Most  Necessary." 
Secure  the  opinion  of  business  men  and  manufacturers. 

4.  Report  on  "Arithmetic  Computation  Tables  in  General  Use." 
Make  clear  their  value  and  range  of  use.  Give  sections  of  typical  tables 
illustrating  their  use  and  the  principles  of  their  constructions. 

5.  Write  a  brief  on  "Machines  for  Arithmetical  Computation." 
Cover  range,  usefulness,  speed,  accuracy,  cost  and  extent  of  use. 


APPENDIX  I. 


SIGNS  AND  SYMBOLS. 


a/c  . . .  .account. 

a/s  ....  account  sales. 

+ addition. 

O.K.  .  .  all  correct. 

& and. 

@ at,  each,  to. 

and  so  on. 

B/L  ...bill  of  lading. 

c/o care  of. 

V check  mark,  correct. 

0 circle. 

c cent. 

° degree. 

-^ division. 

$ dollar. 

= equal,  equals. 

o equivalent. 

' foot,  minutes. 

.fourths  (written,  as 
exponents:  2^  =  2i) 


1.  2.  3 


> greater  than. 

C hundred. 

" inches,  seconds. 

< less  than. 

X multiplication;  incor- 
rect. 

# number  (written  before 

a  figure). 

o/d ....  on  demand. 

% per  centum. 

# pounds  (written  after 

a  figure). 

£ pound  sterling. 

: ratio. 

*.' since. 

>/". . .  .square  root  of. 

— subtraction. 

.' therefore. 

M thousand. 

A triangle. 


419 


APPENDIX  II. 


STANDARD  ABBRJSVIATIONS. 

Note.     The  singular  form  is  now  commonly  used  for  singu- 
lar and  plural,  unless  otherwise  noted. 


Ac Account. 

A acre. 

agt agent. 

ans answer. 

B.O back  order. 

bg bag. 

bal balance. 

bl bale. 

bk. ......  bank,  book. 

bbl barrel. 

bkt basket. 

bot bought. 

bx box. 

bu bushel. 

en can. 

cd card,  cord. 

car carton. 

cs case. 

C,  C.B.  .  .  cash,  cash  book. 

Cash cashier. 

csk cask. 

c,  ct cent. 

eg centigram. 

cm. centimeter. 

ck check. 

ch chest. 


c.o.d collect  on  delivery. 

cml .commercial. 

com commission. 

Co company. 

consgt consignment. 

cr crate,  credit, 

creditor. 

cu cubic. 

cwt hundredweight. 

da day. 

dr debt,  debtor,  debit. 

dept department. 

do ditto  (the  same). 

dol dollar. 

dz.,  doz.  .  .  dozen. 

dft draft. 

ea each. 

E.O.E errors     and     omis- 
sion excepted. 

etc et    cetera    (and    so 

forth). 

ex example,  express. 

exch exchange. 

e.  g exempli  gratia  (for 

example). 

exp expense. 

420 


STANDARD  ABBREVIATIONS. 


421 


far farthings. 

fir firkins. 

f ol folio,  page. 

ft foot. 

for foreign. 

fw.,  fwd.  .  .forward. 

f  r franc. 

f.o.b.  .  .  .  .  .free  on  board. 

gal gallon. 

gi gill- 

g gram. 

gr gross,  grain. 

guar guaranty,  guar- 
antee. 

hf half. 

hhd hogshead. 

hr hour. 

ewt hundredweight. 

in inch. 

int interest. 

I,  inv invoice. 

invt inventory. 

ins insurance. 

inst instant  (the  pres- 
ent month). 

kg keg. 

km kilometer. 

L.F ledger  folio. 

l.p list  price. 

If.,  Iv loaf,  loaves. 

lb pound. 

m.p marked  price. 

mdse merchandise. 

m meter. 


Messrs.  .  . .  Messieurs  (gentle- 
men. Sirs). 

mi mile. 

min minute. 

Mr Mister. 

Mrs Mistress. 

mo month. 

no number. 

O.B order  book. 

oz ounce. 

p.,  PP page,  pages. 

pkg package. 

pd paid. 

pi pail. 

pr pair. 

pay payment. 

d pence. 

per per  (by  the). 

p.c per  centum, 

pk peck. 

pwt pennyweight. 

pc piece. 

pt pint. 

poc,  pkt.  .  pocket. 

lb pound. 

Pres President. 

prox proximo    (the    fol- 
lowing month). 

qt quart. 

qr quire. 

R.R railroad. 

rm ream. 

recM received. 

M Reischsmark. 


422 


BUSINESS  ARITHMETIC. 


rd rod. 

sk sack. 

S sales. 

sec second. 

sec'y- ....  secretary 

s shilling. 

set settlement. 

ship shipment. 

shipt shipped. 

sig signed,  signature. 

st street. 

stk stock. 

sund sundries. 

trc .tierce. 


T ton. 

tr.,  trans.  .  transfer. 

treas treasurer,  treasury. 

tb tub. 

ult ultimate  (last 

month). 

via via  (by  way  of). 

viz videlicet  (namely,  to 

wit). 

wk week. 

wt weight. 

yd yard. 

yr year. 


INDEX. 


Abbreviations,  420-422. 
Acceptance,  309-311. 
Accident  insurance,  257. 
Account,  72. 

averaging,  397. 

cash,  73. 

expense,  74. 

merchandise,  73. 

personal,  74. 

proprietor's,  75. 
Account  sales,  249. 
Accuracy,  7. 
Accurate  interest,  293. 
Acre  foot,  177. 
Acute  angle,  143. 
Addition,  7. 

checking,  11. 

complements,  10. 

dictation,  7. 

grouping,  9. 

horizontal,  12. 

of  decimals,  15. 

of  denominate  numbers,  138. 

of  fractions,  95. 

principles  of,  8-12. 
Ad  valorem  duties,  278,  283. 
Advertising,  77. 

newspaper,  periodical,  79,  80. 

billboard,  79. 

poster,  78,  79. 

records  of  replies,  83,  84,  85. 
Agency,  245. 
Agent,  245. 

Agent,  payment  of,  246. 
Aliquot  parts,  110. 

division  by^  115. 

multiplication  by,  110. 
Altitude,  145. 
Analysis  of  problems,  118. 
Angle,  143. 

Angular  measure,  134. 
Antecedent,  186. 
Appraiser,  279. 


Approximate  results,  40. 

time,  169. 
Arabic  notation,  3. 
Arc,  145. 
Area  of  circle,  147. 

of  parallelogram,  146. 

of  triangle,  146. 
Assav  office,  51. 
Assessment,  269,  270,  274,  389. 
Assessors,  270. 
Automobile  insurance,  257. 
Average,  43. 
Average  clause,  264. 
Average  investment,  394. 
Average  storage,  416. 
Averaging,  43,  397. 
Avoirdupois  weight,  133. 
Axioms,  46. 

B 

Bank  discount,  318. 

register,  325. 

rules  for,  320 

terms  of,  319. 

ticket,  324. 
Bank  draft,  361,  362. 
Banker's  60-day  method,  287. 
Base,  145. 

Base  for  percentage,  215,  221. 
Base  for  profit  and  loss,  226. 
Base  ball  averages,  219. 
Bear,  344. 

Beneficiary,  257,  259. 
Bids  and  estimates,  383. 
Bill,  discount,  409. 

manufacturers,  407,  409. 

monthly  statement,  410. 

professional,  69. 

simple,  403,  405. 

storage,  415. 
Billing,  403. 

Bill  of  exchange,  370,  371. 
Bill  of  lading,  280. 
Bin,  capacity  of,  162. 


423 


424 


INDEX. 


Blank  endorsement,  307. 
Board  foot,  161. 
Boiling  point,  175. 
Bond,  309,  337. 

compared  with  stock,  337. 

coupon,  337. 

quotations,  339. 

registered,  337. 
Bonded  warehouse,  280. 
Bonus,  69. 
Broker,  245. 
Brokerage,  245. 
Building  and  loan  associations,  351. 

distribution  of  profits,  353. 

earnings,  351. 

loans,  351. 
Bull,  344. 

Business  accounts,  72. 
Business  terms,  72. 


Calculation  of  time  table,  170-171. 
Call  loans,  345. 
Canadian  money,  135. 
Cancellation,  88,  286. 
Capacity,  measures  of,  133. 

of  bins,  tanks,  etc.,  162. 
Capital,  334. 
Capital  stock,  332. 
Carat,  133. 
Carpeting,  155. 
Cash  balance,  402. 
Cashier's  check,  361. 
Casting  out  nines,  11. 
Centigrade  scale,  175. 
Certificates,  gold  and  silver,  50. 
Certificates  of  deposit,  362. 
Change  schedule,  66. 
Check,  357,  358. 
Checking  the  work,  11,  118. 
Cu-cle,  145. 

Circular  measure,  134. 
Circulation  of  money,  62. 
Circulation  statement,  12. 
Circumference,  145. 
Coal  consumption,  210. 
Coins  of  the  United  States,  51. 
Cold  storage  temperatures,  176. 
Collateral,  345. 

loan,  345. 

note,  346. 
Collection  fee,  361. 


Collector  of  the  port,  279. 
Commercial  discount,  236. 
Commercial  draft,  363. 
Commission  and  brokerage,  245. 
Commission  merchant,  245. 
Common  stock,  334. 
Complements,  10. 
Composite  measures,  177. 
Composite  numbers,  86. 
Compound  interest,  302. 

tables,  303. 
Compound  proportion,  190. 
Compounds,  223. 
Cone,  156. 
Conical  surface,  156. 
Consequent,  186. 
Consideration,  255. 
Consignee,  246. 
Consignment,  246. 
Consignor,  246. 
Contract,  255. 

Conversion  of  fractions,  103. 
Conversion  tables,  184. 
Coordinates,  200. 
Copying  numbers,  8. 
Cord  measure,  160. 
Com,  production  of,  205. 
Corporation,  332. 

advantages  of,  333. 
Correspondent  bank,  360. 
Cost,  226,  230. 

Cost  by  hundred  and  thousand,  116. 
Cost  per  inquiry,  83. 
Cost-keeping,  379. 

ticket,  380,  381. 

unit  of,  382. 
Cotton,  production  of,  209. 
Counting  table,  134. 
Coupon,  337. 
Credit  insurance,  258. 
Credit  prices,  295. 
Cube,  156. 

Cube  of  numbers,  121. 
Cube  root,  125. 

graphic  illustration,  129. 
Cubic  measure,  135. 

metric,  183. 
Curb  market,  343. 
Currency  memorandum,  66. 
Currency  of  U.  S.,  50. 
Custom  house,  279. 
Customs,  277. 
Cylinder,  156. 


INDEX. 


425 


Date  line,  167. 

Date  of  maturity,  172. 

Decimals, 

addition  of,  15. 

division  of,  39. 

multiplication  of,  31. 

subtraction  of,  22. 
Degree,  circular  measure,  165. 
Denominate  numbers,  132. 

addition,  138. 

division,  141. 

multiplication,  140. 

reduction  ascending,  137. 

reduction  descending,  136. 

subtraction,  139. 

tables  of,  53,  133-135. 
Denominator,  91. 
Deposit,  359. 
Depreciation,  375. 

on  original  values,  375. 

on  reduced  values,  375. 

table  of,  377. 
Diagram,  193. 
Diameter,  145. 
Dictation,  7. 
Difiference  column,  20. 
Dimensioning,  195. 
Discount,  236. 

bank,  318. 

rates,  240. 

series,  238. 
Discussion — See  Questions  for  dis- 
cussion. 
Distribution  of  profits,  389-395. 
Dividend,  25,  335. 
Divisibility,  tests,  86,  87. 
Division,  35. 

by  continued  subtraction,  35. 

long,  37. 

of  decimals,  39. 

of  denominate  numbers,  141. 

of  fractions,  101. 

short,  36. 

short  methods  of,  38. 
Domestic  exchange,  355. 
Draft,  309. 
Drawee,  309. 
Drawer,  309. 
Dry  measure,  133. 
Duties,  ad  valorem,  277,  283. 

specific,  277,  282. 


£ 

Endowment  policy,  259. 
Endorsement, 

blank,  307. 

fuU,  308. 

quahfied,  308. 
English  money,  135. 
Equality  of  expressions,  46. 
Equated  date,  397. 
Equation,  46. 

Equation  of  accounts,  397,  399. 
Equation  of  payments,  397. 
Equilateral  triangle,  144. 
Estimates,  383. 
Evolution,  122. 
Exact  interest,  293. 
Exact  results,  40. 
Exact  time,  169. 
Exchange,  355. 

domestic,  355. 

fluctuation  of,  364,  371. 

foreign,  366. 

inland,  355. 

rates,  364,  366,  373. 
Ex-dividend,  343. 
Expenditures  of  U.  S.  Gov't,  277. 
Exponent,  121. 
Exports  of  U.  S.,  208,  219. 
Express  money  order,  356. 
Expressions,  equality  of,  46. 
Extension,  measures  of,  134. 
Extremes,  190. 


F 

Face,  156. 

Factor,  agent,  245. 

Factors  and  multiples,  86. 

Factor,  highest  common,  88. 

Factoring,  87. 

Factory  expense,  68. 

Fahrenheit  (temperature),  175. 

Fee,  269. 

Fidelity  insurance,  257. 

Figures,  geometric,  143. 

Fire  insurance,  265. 

Flooring,  151. 

Floor  plan,  152. 

Focal  date,  398. 

Foot  pound,  177. 

Foreign  coin,  gold  value  of,  281. 

Foreign  exchange,  366. 


426 


INDEX. 


Foreign  money  order,  368. 
Formula,  179. 
Fractions,  91-109. 

addition,  95. 

compound,  92. 

conversion  of,  103. 

division  of,  101. 

improper,  92. 

lowest  terms,  93. 

multiplication  of,  97. 

proper,  92. 

reduction  of,  91. 

similar,  95. 

simplification  of,  93. 

subtraction  of,  96. 

terms,  91. 
Fractional  relations,  106. 
Fractional  units,  91. 
Free  list,  278. 
Freezing  point,  175. 
French  money,  135. 


Gable,  151. 

Gain,  72. 

Geometric    conceptions,     143-147: 

156-160. 
Geometric  line,  143. 
German  money,  135. 
Gold  value  of  coin,  281. 
Good  will,  388. 
Graph,  193;  204-211. 
Graphic  arithmetic,  193. 
Graphic  treatment  of  squares  and 

cubes,  128. 
Greatest     common     divisor  —  See 

Highest  common  factor. 
Gross  cost,  245. 
Gross  proceeds,  246. 
Grouping,  in  addition,  9 
Guaranty,  246. 


Hundreds,  cost  of,  116. 
Hypotenuse,  149. 


Immigration,  table  of,  210. 

Import  duties,  277. 

Imports  of  U.  S.,  211. 

Improper  fraction,  92. 

Income,  347. 

Income  insurance,  262. 

Individual  work,  42,  142,  163,  174, 

185,  212,  219,  235,  324,  412, 

417,  418. 
Indorsement — See  Endorsement. 
Insolvency,  388. 
Insurance,  255. 

companv,  259,  264. 

personal,  256-263. 

poUcy,  256,  259. 

property,  256;  263-268. 
Interest, 

accurate,  293. 

banker's  60-day  method,  287. 

cancellation  method,  285. 

compound,  302. 

simple,  285. 

six  per  cent,  method,  290. 

tables,  292,  303. 
Interest  on  investments,  392. 
Investment,  347. 
Invoices,  279,  282,  283. 
Involution,  121. 
Irrigation,  206. 
Isosceles  triangle,  144. 


Job  work  ticket,  380. 


Knot,  134. 


Health  insurance,  257. 
Hemisphere,  157. 
Highest  common  factor,  88. 
Holidays,  313. 
Horizontal  addition,  12. 
Horsepower,  177. 
Household  expense,  21. 
How  to  read  numbers,  4. 


Land  measure, 

metric,  183. 

surveyor's,  134. 
Lateral  area,  159. 
Lawful  money,  50. 
Least  common  multiple,  89. 
Legal  rate  of  interest,  285. 
Letter  of  credit,  369,  370. 
Lever,  188. 


INDEX. 


427 


Liabilities,  72. 
Liability  insurance,  257. 
License,  269,  274. 
Life  insurance,  259. 

annual  premiums,  261. 

values,  260. 
Limited  partner,  388. 
Linear  measure,  134. 
Linear  metric  measure,  182. 
Linear  scale,  195. 
Lines,  143. 
Liquidation,  338. 
Liquid  measure,  133. 
Loans,  306,  345. 
Long  division,  37. 
Longitude,  165. 
Loss,  72. 
Lumber,  160. 


Maker  of  note,  307. 
Making  change,  54. 
Map  length,  202,  203. 
Margin,  343. 
Marking  goods,  232. 
Marine  insurance,  265. 
Maturity,  172. 

amount  due  at,  319. 

of  negotiable  instruments,  312. 
Mean,  43. 
Means,  190. 
Measure, 

angular,  134. 

apothecaries',  133. 

avoirdupois,  133. 

circular,  134. 

comparative,  133. 

composite,  177. 

cord,  160. 

counting,  134. 

cubic,  135. 

dry,  133. 

extension,  134. 

linear,  135. 

liquid,  133. 

long,  134. 

metric,  182. 

money,  135. 

sea,  134. 

square,  134. 

surveyor's,  134. 

time,  135,  162. 


Measure, 

temperature,  175. 

Troy,  133. 

value,  135. 

weight,  133. 
Measuring,  136. 
Mensuration,  143-163. 
Merchant's  rule,  317. 
Meridian,  164, 
Metric  equivalents,  184. 
Metric  system,  182. 
Mexican  money,  135. 
Mints,  51. 
Minuend,  18. 
Mixed  numbers,  92. 

division  of,  101. 

multiplication  of,  99. 
Mixtures,  223. 
Money, 

circulation  of  U.  S.,  52. 

recoinage  of  U.  S.,  52. 

table  of  foreign,  135. 

table  of  U.  S.,  53. 

United  States,  49. 
Money  order,  355. 

form  of  express,  356. 

rates,  356,  368. 
Multiple,  86. 

common,  89. 
Multiplication,  25. 

aliquot  parts,  25. 

commutative  law,  25. 

decimal,  31. 

denominate,  140. 

fractional,  97. 

shortening,  28-30. 

tables,  26. 
Multiplier,  25. 


Negotiable  paper,  307. 

bank  draft,  361,  362. 

certificate  of  deposit,  362. 

check,  357. 

maturity,  312. 

promissory  note,  307. 

sight  draft,  309. 

time  draft,  310,  311. 
Net  capital,  72,  388. 
Net  cost,  246. 
Net  gain,  72. 
Net  insolvency,  388. 


428 


INDEX. 


Net  loss,  72. 

Net  proceeds,  246. 

Newspaper, 

advertising,  79. 

advertising  rates,  81. 

circulation,  12,  80. 
Nines,  casting  out,  11. 
Nominal  partner,  388. 
Normal,  43. 
Notation, 

Arabic,  3. 

Roman,  5. 
Notes,  307,  320. 
Notice  of  protest,  308. 
Numbers,  1. 

abstract,  1. 

composite,  86. 

compound,  2. 

concrete,  1. 

copying,  8. 

denominate,  1,  132. 

dictation  of,  7. 

like,  1. 

powers  of,  121. 

prime,  86. 

roots  of,  122. 

simple,  2. 

unlike,  1. 
Numeration,  4,  5. 
Numerator,  91. 


Obtuse  angle,  143. 
Old  age  insurance,  262. 
Open  policy,  264. 


Panama  canal,  223. 
Papering,  153. 
Paper  table,  135. 
Parallel  line  graphs,  207. 
Parallel  lines,  143. 
Parallelogram,  144. 
Parcels  post,  57,  58. 
Partial  payments,  314. 

indorsements,  315. 

merchant's  rule,  317. 

United  States  rule,  315. 
Partition,  36. 
Partitive  proportion,  387. 
Partners,  388. 
Partnership,  387. 


Partnership  statement,  392. 
Par  value,  333. 
Passenger  mile,  178. 
Payee,  307. 
Payments,  61,  306. 
Pay  roll,  63,  65. 
Pay  roll  check,  358. 
Percentage,  213. 

comparative  table,  214. 

to  find  base,  221. 

to  find  percentage,  215. 

to  find  rate,  217. 
Perimeter,  145. 
Periodic  interest,  300. 
Perpendicular  line,  143. 
Pitch  of  a  roof,  151. 
Plane  figures,  143. 
Plane  surfaces,  143. 
Plastering,  153. 
Plotting,  194,  197,  199. 
Policy  of  insurance,  256. 

open,  264. 

valued,  264. 
Poll  tax,  270. 
Polyhedron,  156. 
Port  of  delivery,  279. 
Port  of  entry,  279.      . 
Postage,  56. 

report,  59,  60. 

stamps,  57. 
Postal  information,  57. 
Postal  money  order,  356. 
Post  oflBces  and  routes,  56. 
Posters,  78,  79. 
Power  of  a  number,  121. 
Practical  measurements,  143-175. 
Precipitation  chart,  209. 
Preferred  stock,  334. 
Premium,  256. 

rates  of  insurance,  261. 

wage,  69. 
Present  worth  and  true  discount, 

299. 
Present  worth, 

of  a  debt,  299. 

of  a  firm,  72,  388. 
Price  per  hundred  or  thousand,  116, 
Prime  cost — See  Net  cost. 
Prime  factor^  86. 
Prime  meridian,  164. 
Prime  number,  86. 
Prmcipal,  285. 
Prism,  156. 


INDEX. 


429 


Problem  analysis  and  solution,  118. 
Proceeds  of  a  note,  320. 
Proceeds  of  a  sale,  246. 
Production  of  barley,  193. 
Profit  as  rate  of  selling  price,  228. 
Profit  and  loss,  225. 
Profit  and  loss  account,  390 
Profit  sharing,  SSO-391. 
Promissory  note,  307. 
Proper  fraction,  92. 
Property  tax,  269. 
Proportion,  190. 

compound,  190. 

continued,  190. 

direct,  191. 

indirect,  191. 

partitive,  387. 

principles  of,  190. 

simple,  190. 

terms,  190. 
Proprietor's  account,  75. 
Proposal,  383. 
Protest,  388. 
Purchase,  249. 
Pyramid,  156. 


Quadrilateral,  144. 
Qualified  endorsement,  308. 
Questions  for  discussion,  55,  90,  105, 

195,  215,  232,  238,  313,  318, 

319,  388,  396. 
Quotation,  334,  338. 
Quotation  list,  251,  339. 
Quotient,  35. 


Radius,  145. 

Rate,  213,  217. 

Rate  card,  81. 

Rate  of  exchange,  364,  366. 

fluctuation  of,  364,  371. 
Rate  of  interest,  285. 
Ratio,  186. 

direct,  186. 

inverse,  186. 

signs  of,  186. 

terms,  186. 

value,  186. 
Ration  list,  27. 

Reading  and  writing  numbers,  3-5. 
Reaumur  thermometer,  75. 


Receipts  of  U.  S.  gov't,  277. 
Receipts  for  payment,  74,  404. 
Reciprocal  of  a  fraction,  101. 
Record, 

of  advertising,  84,  85. 

of  postage,  59,  60. 

of  professional  visits,  71. 
Rectangle,  145. 
Rectangular  solid,  156. 
Reduction  of  denominate  numbers, 

136,  137. 
Reduction  of  plotted  lengths,  197. 
Registered  bond,  337. 
Relations,  fractional,  106. 
Remainders,  35. 
Requisition,  411. 
Requisition  book,  412. 
Resistance,  192. 

Results,  exact  and  approximate,  40. 
Right  angle,  143. 
Right  angled  triangle,  144. 
Roman  notation,  5. 
Root  of  numbers,  122. 


S 

Salary,  62. 

Salary  pay  roll,  63. 

Sales  (agent),  247. 

Savings  accounts,  326,  328, 329, 331., 

interest  on,  328. 
Savings  banks,  326. 
Scalene  triangle,  144. 
Scales,  194. 
Sea  measure,  134. 
Selling  price,  226. 
Services, 

biU  for,  69. 

payment  for,  61. 
Shareholder,  332. 
Short  division,  36. 
Short  methods, 

multiphcation,  28,  30,  100. 

division,  38. 
Sight  draft,  309,  361. 
Signs,  algebraic  and  numerical,  47- 
Simple  interest,  285. 
Sinking  fund,  337,  377,  378. 
Six  per  cent,  method,  290. 
Sixty-day  method,  287. 
Solids,  156. 

Solution  of  problems,  118 
Solving  equations,  45. 


430 


INDEX. 


Specifications,  383. 
Specific  duties,  277,  282. 
Specific  gravity,  41. 
Sphere,  157. 
Spherical  surface,  156. 
Square,  144. 

Square  (lumber  measure),  151. 
Square  measure,  134. 

metric,  183. 
Square  of  a  number,  121. 

graphic  illustration,  128. 
Square  root,  123. 

graphic,  128. 
Standard  time,  167. 
Standard  measures,  133-135. 
Standard  weights,  133-135. 
Statement,  monthly,  410. 

Statistics!  14^  193-211. 
Stock  broker,  343. 
Stock  certificate,  333. 
Stockholder,  332. 
Stock  dividends,  336. 
Stock  record,  20. 
Stock,  332. 

buying  and  selling,  342. 

classes  of,  334. 

quotations,  334,  338. 

shares,  333. 
Storage,  414. 

average,  416. 

bill,  415. 

term,  414. 
Subtraction,  18. 

checking,  18. 

complement  method,  19. 

of  decimals,  22. 

of  denominate  numbers,  139. 

of  fractions,  96. 
Subtrahend,  18. 
Surplus,  335. 
Surveyor,  280. 
Surveyor's  measure,  134. 
Symbols,  419. 


Tables, 

apothecaries'  weight,  133. 
avoirdupois  weight,  133. 
Canadian  money,  135. 
circular  measure,  134. 
circular  measure  and  time,  165. 
conversion,  133. 


Tables, 

counting,  133. 

cubic  measure,  135. 

denominate  numbers,  133-35. 

dry  measure,  133. 

English  money,  135. 

French  money,  135. 

German  money,  135. 

import  duties,  278. 

insurance  premiums,  261. 

interest,  292. 

linear  measure,  134. 

liquid  measure,  133. 
.  metric,  182. 

metrical  equivalents,  184. 

Mexican  money,  135. 

multiplication,  26. 

paper,  134. 

siiJcing  fund,  378. 

square  measure,  135. 

tax,  273. 

time,  137,  170,  171. 

Troy  weight,  133. 

United  States  money,  53. 

values,  insurance,  260. 

wage,  67. 
Tabulations,  14. 
Tanks,  capacity  of,  162. 
Tariff,  277. 
Tariff  act,  277. 
Taxation,  269. 
Tax  table,  273. 
Telegraph  money  order,  357. 
Temperature,  175. 

scale  reduction,  176. 
Temperature  chart,  209. 
Term  of  discount,  319. 
Term  of  note,  319. 
Terms  of  a  fraction,  91. 
Terms  of  a  sale.  405. 
Tests  of  divisibihty,  86,  87. 
Textile  manufactures,  208. 
Thousands,  to  find  cost  of,  116. 
Timber  supply,  207. 
Time, 

calculation  table,  170-171. 

commercial,  168. 

difference  in,  166. 

standard,  167. 
Time  card,  68. 
Time  draft,  309-311. 
Time  of  discount,  119 
Time  sheet,  68. 


INDEX. 


431 


Ton  mile,  178. 
Train  mile,  178. 
Transit  insurance,  266. 
Transportation,  47. 
Travelers'  check,  368. 
Triangle,  144. 
Troy  weight,  133. 
True  discount,  299. 
Turn-over  of  capital,  227. 
Types  of  problems,  118, 119. 


Value  of  foreign  coin,  281. 
Valued  policy,  264. 
Visits,  record  of,  71. 


Volume, 


cylinder,  159. 
prism,  157. 


Undivided  profits,  335. 
Unit, 

of  freight  traffic,  178. 

of  irrigation,  177. 

of  passenger  traffic,  178. 

of  power,  177. 

of  train  traffic,  178. 

of  work,  177. 
United  States  customs,  275. 
United  States  money,  49,  50. 
United  States  rule,  315. 
Units,  1. 

fractional,  91. 
Unity,  1. 
Usury,  285. 


Wages,  62,  64. 
Wage  earners,  61. 
Wage  pay  roll,  65. 
Wage  table,  67. 
Wall  signs,  78. 
Warehouse,  280. 
Watered  stock,  325. 
Weight, 

apothecaries',  133. 

avoirdupois,  133. 

comparison  of,  133. 

long  ton,  133. 

miscellaneous,  133. 

metric,  183. 

Troy,  133. 
Whole  life  policies,  259. 


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